user3237: Technically it is possible for a game to last more than 24 hours, since after 40 or 60 moves each player generally gets a 30-second increment.
In an endgame, the player with the advantage will just need to be careful of the fifty-move rule and the threefold repetition rule, and he would be able to keep pressing for many many hours hoping that his opponent will crack and make a mistake.
But the problem is: 24 hours after the game has started, the next round would need to start!
So what would happen?
Some possibilities occur to me, but I don't know which (if any) might apply:

The entire round might be delayed for everyone participating in the event;
the next round might start normally on time for everyone, except for 4 players (including the 2 players playing the problematic game) who would need to wait for the problematic game to end;
the next round might start normally on time for everyone, and the two players who are still playing the problematic game would need to play their next round games while simultaneously continuing the problematic game;
the arbiters might simply decide to end the problematic game, considering it as a draw;
the FIDE President might need to add a new rule to prevent this kind of situation from happening again.


user4418: 
Technically it is possible for a game to last more than 24 hours

Not true in a game where both players are trying to win or draw. You haven't done your sums.

In an endgame, the player with the advantage will just need to be
  careful of the fifty-move rule

In 20 hours of 30 second increment play at least 1200 moves have to be played. That is at least 24 times when there is a flurry of captures or pawn moves. I say "flurry" because generally speaking when there is a capture there is also a recapture. There are only 16 pieces and 16 pawns on the board when the game starts and after the first 4 hours of play a lot of them will be gone. 

user8160: Would the entire round be delayed for everyone participating in the event?
No, if the game duration may exceed a reasonable amount of time (I am not talking about 24 hours, but lets say 12 hours or so) then the game is adjourned and any tournament director would plan a special day for adjourned games. This works perfectly in case of so called Round-robin system tournament. In case if the tournament is played by Swiss system rules then it is simply reasonable to have shorter time limits per game, so all games will end by the end of the next round.
Would the arbiters simply decide to end the problematic game, considering it as a draw?
No, this could never happen! The game can be considered a draw in case, for example one side loses on time while the other side doesn't have sufficient material to mate, i.e. two knights only, etc)
Would the FIDE President need to add a new rule to prevent this kind of situation from happening again?
There is no way to regulate all possible outcomes of bad tournament planning. It's not FIDE responsibility. 

user2001: Well, a game that lasts more than 8 hours would make everyone involved in the tournament tired and angry. Under the assumption of today's normal time controls with 90 minutes for 40 moves followed by 30 additional minutes for the rest of the game and 30 seconds increment per move starting from move one.
It is an impressive effort to reach 24 hours of gameplay. The 50-move rule will be crucial. Most probably, 49 moves would be made with the pieces, followed by pushing one pawn one step forward and thereafter repeating the cycle. Both players would have to be in agreement and collaborate to reach this epic achievement. A slight complication is that you have to write down your move, which decreases your increment to perhaps 25 seconds and also makes your hand(s) more tired.
My guess is that the arbiter would declare a draw, 1-0, 0-1 or perhaps even 0-0 if possible, after about 8 hours of play. The grumpy arbiter would thereafter most probably try to disqualify both players from the tournament.
Since such situations are so rare, FIDE will probably make a decision based on the particular situation in question, without creating new rules or changing the existing ones.

user2311: Although replies by Brian Towers and StudentT are the right ones (both players would be disqualified) from a theoretical points of view it would be possible that a game has not ended in time for the next round to begin. I would even say that it's possible that it could actually happen in a tournament with 3 rounds per day.
I case it would happen I guess the game would be adjourned by the arbiters. The entire round would not be delayed/cancelled, but the result would be considered to be a draw for pairing purposes only. See General handling rules for Swiss Tournaments, point D.1, and Laws of Chess, point E.1.a.
From Laws of Chess

If a game is not finished at the end of the time prescribed for play,
  the arbiter shall require the player having the move to ‘seal’ that
  move. The player must write his move in unambiguous notation on his
  scoresheet, put his scoresheet and that of his opponent in an
  envelope, seal the envelope and only then stop the chess clock. Until
  he has stopped the chess clock the player retains the right to change
  his sealed move. If, after being told by the arbiter to seal his move,
  the player makes a move on the chessboard he must write that same move
  on his scoresheet as his sealed move.

from General handling rules for Swiss Tournaments

Adjourned games are considered draws for pairing purposes only.

Update: I've found a real example in a Catalan tournament (Masdenverge). You can read tournament rules here: http://www.escacsmasdenverge.com/. It was an U-2200 tournament (to allow a 60+30 time control) and three rounds per day were scheduled. See the note where this possible problem is mentioned:

NOTA: Si una partida s'allargara en excés, superant l'hora prevista
  per a l'inici de la següent ronda, l'àrbitre principal podrà segons el
  seu criteri, emparellar amb el resultat provisional de taules, a
  l'efecte de que els altres participants puguen jugar la ronda següent.
  Quedaran ajornades les partides dels jugadors que estiguen jugant la
  ronda anterior i una vegada finalitzada, s'unixen a la ronda.

A free English translation:

If a game lasts for too long, exceeding the scheduled time for the
  start of the next round, the main arbiter may at his discretion,
  proceed to the pairing with the interim result of a draw, to the
  effect that the other players can play the next round. Games of the
  player who are still playing the previous round will be delayed and
  once these games end they'll join the current round.

As you can see in this case the draw for pairing purposes rule is used, but games are not adjourned but the players will continue to the end, and then they'll join the next round.

3237: <p>Technically it is possible for a game to last more than 24 hours, since after 40 or 60 moves each player generally gets a 30-second increment.</p>
<p>In an endgame, the player with the advantage will just need to be careful of the fifty-move rule and the threefold repetition rule, and he would be able to keep pressing for many many hours hoping that his opponent will crack and make a mistake.</p>
<p>But the problem is: 24 hours after the game has started, the next round would need to start!</p>
<p><strong>So what would happen?</strong></p>
<p>Some possibilities occur to me, but I don't know which (if any) might apply:</p>
<ul>
<li>The entire round might be delayed for everyone participating in the event;</li>
<li>the next round might start normally on time for everyone, except for 4 players (including the 2 players playing the problematic game) who would need to wait for the problematic game to end;</li>
<li>the next round might start normally on time for everyone, and the two players who are still playing the problematic game would need to play their next round games while simultaneously continuing the problematic game;</li>
<li>the arbiters might simply decide to end the problematic game, considering it as a draw;</li>
<li>the FIDE President might need to add a new rule to prevent this kind of situation from happening again.</li>
</ul>
 4418: <blockquote>
  <p>Technically it is possible for a game to last more than 24 hours</p>
</blockquote>

<p>Not true in a game where both players are trying to win or draw. You haven't done your sums.</p>

<blockquote>
  <p>In an endgame, the player with the advantage will just need to be
  careful of the fifty-move rule</p>
</blockquote>

<p>In 20 hours of 30 second increment play at least 1200 moves have to be played. That is at least 24 times when there is a flurry of captures or pawn moves. I say "flurry" because generally speaking when there is a capture there is also a recapture. There are only 16 pieces and 16 pawns on the board when the game starts and after the first 4 hours of play a lot of them will be gone. </p>
 8160: <p>Would the entire round be delayed for everyone participating in the event?</p>

<p>No, if the game duration may exceed a reasonable amount of time (I am not talking about 24 hours, but lets say 12 hours or so) then the game is adjourned and any tournament director would plan a special day for adjourned games. This works perfectly in case of so called Round-robin system tournament. In case if the tournament is played by Swiss system rules then it is simply reasonable to have shorter time limits per game, so all games will end by the end of the next round.</p>

<p>Would the arbiters simply decide to end the problematic game, considering it as a draw?</p>

<p>No, this could never happen! The game can be considered a draw in case, for example one side loses on time while the other side doesn't have sufficient material to mate, i.e. two knights only, etc)</p>

<p>Would the FIDE President need to add a new rule to prevent this kind of situation from happening again?</p>

<p>There is no way to regulate all possible outcomes of bad tournament planning. It's not FIDE responsibility. </p>
 2001: <p>Well, a game that lasts more than 8 hours would make everyone involved in the tournament tired and angry. Under the assumption of today's normal time controls with 90 minutes for 40 moves followed by 30 additional minutes for the rest of the game and 30 seconds increment per move starting from move one.</p>

<p>It is an impressive effort to reach 24 hours of gameplay. The 50-move rule will be crucial. Most probably, 49 moves would be made with the pieces, followed by pushing one pawn one step forward and thereafter repeating the cycle. Both players would have to be in agreement and collaborate to reach this epic achievement. A slight complication is that you have to write down your move, which decreases your increment to perhaps 25 seconds and also makes your hand(s) more tired.</p>

<p>My guess is that the arbiter would declare a draw, 1-0, 0-1 or perhaps even 0-0 if possible, after about 8 hours of play. The grumpy arbiter would thereafter most probably try to disqualify both players from the tournament.</p>

<p>Since such situations are so rare, FIDE will probably make a decision based on the particular situation in question, without creating new rules or changing the existing ones.</p>
 2311: <p>Although replies by Brian Towers and StudentT are the right ones (both players would be disqualified) from a theoretical points of view it would be possible that a game has not ended in time for the next round to begin. I would even say that it's possible that it could actually happen in a tournament with 3 rounds per day.</p>

<p>I case it would happen I guess the game would be adjourned by the arbiters. The entire round would not be delayed/cancelled, but the result would be considered to be a draw for pairing purposes only. See <a href="https://www.fide.com/fide/handbook.html?id=84&amp;view=article" rel="nofollow noreferrer">General handling rules for Swiss Tournaments</a>, point D.1, and <a href="https://www.fide.com/fide/handbook.html?id=171&amp;view=article" rel="nofollow noreferrer">Laws of Chess</a>, point E.1.a.</p>

<p>From Laws of Chess</p>

<blockquote>
  <p>If a game is not finished at the end of the time prescribed for play,
  the arbiter shall require the player having the move to ‘seal’ that
  move. The player must write his move in unambiguous notation on his
  scoresheet, put his scoresheet and that of his opponent in an
  envelope, seal the envelope and only then stop the chess clock. Until
  he has stopped the chess clock the player retains the right to change
  his sealed move. If, after being told by the arbiter to seal his move,
  the player makes a move on the chessboard he must write that same move
  on his scoresheet as his sealed move.</p>
</blockquote>

<p>from General handling rules for Swiss Tournaments</p>

<blockquote>
  <p>Adjourned games are considered draws for pairing purposes only.</p>
</blockquote>

<p>Update: I've found a real example in a Catalan tournament (Masdenverge). You can read tournament rules here: <a href="http://www.escacsmasdenverge.com/" rel="nofollow noreferrer">http://www.escacsmasdenverge.com/</a>. It was an U-2200 tournament (to allow a 60+30 time control) and three rounds per day were scheduled. See the note where this possible problem is mentioned:</p>

<blockquote>
  <p>NOTA: Si una partida s'allargara en excés, superant l'hora prevista
  per a l'inici de la següent ronda, l'àrbitre principal podrà segons el
  seu criteri, emparellar amb el resultat provisional de taules, a
  l'efecte de que els altres participants puguen jugar la ronda següent.
  Quedaran ajornades les partides dels jugadors que estiguen jugant la
  ronda anterior i una vegada finalitzada, s'unixen a la ronda.</p>
</blockquote>

<p>A free English translation:</p>

<blockquote>
  <p>If a game lasts for too long, exceeding the scheduled time for the
  start of the next round, the main arbiter may at his discretion,
  proceed to the pairing with the interim result of a draw, to the
  effect that the other players can play the next round. Games of the
  player who are still playing the previous round will be delayed and
  once these games end they'll join the current round.</p>
</blockquote>

<p>As you can see in this case the draw for pairing purposes rule is used, but games are not adjourned but the players will continue to the end, and then they'll join the next round.</p>
