user159: In competition blitz games, making an illegal move and then pressing the clock forfeits the game if the opponent claims the illegal move as long as they do not make a new move, as a new move would void the previous illegal move according to FIDE blitz rules.
I've seen all kind of forfeits because of this reason in blitz games, including one player taking one of his own pawns.
However, the most common case for illegal moves is neglecting that the King is in check or unprotecting the King. In this case where the King is checked, if the opponent takes the exposed king and presses the clock, are they making a new illegal move, hence voiding the previous one?
I've been told that taking the King, even as a kind of "joke" to expose a previous illegal move, is an illegal move by itself, because the King cannot be taken, only mated. Is this true under FIDE rules?

user12: This is actually very common in blitz games.  Taking the opponent's king is considered to be a claim of an illegal move.  You can see my answer to this question for some more information.
Since taking the opponent's king is a claim instead of a move, it immediately ends the game.  In blitz games, the player "capturing" the king wins.  In slower games, taking the king is in fact illegal, and if a player did this, the position would be reset to the position before the first illegal move, and then any touch move rules would apply.  If there were time pressure considerations, one or both players could have two minutes added to their clocks at the arbiter's discretion.  
From the FIDE Appendix for Blitz and Rapidplay
Article A.4.c:  

An illegal move is completed once the opponent’s clock has been
  started. The opponent is entitled to claim a win before he has made
  his own move. However, if the opponent cannot checkmate the player’s
  king by any possible series of legal moves, then the claimant is
  entitled to claim a draw before he has made his own move. Once the
  opponent has made his own move, an illegal move cannot be corrected
  unless mutually agreed without intervention of an arbiter.


user103: Capturing the King is an illegal move according to FIDE.
Refer to Article 1.2 from the Laws of Chess

The objective of each player is to place the opponent’s king ‘under
  attack’ in such a way that the opponent has no legal move. The player
  who achieves this goal is said to have ‘checkmated’ the opponent’s
  king and to have won the game. Leaving one’s own king under attack,
  exposing one’s own king to attack and also ’capturing’ the opponent’s
  king are not allowed. The opponent whose king has been checkmated has
  lost the game.


159: <p>In competition blitz games, making an illegal move and then pressing the clock forfeits the game if the opponent claims the illegal move as long as they do not make a new move, as a new move would void the previous illegal move according to FIDE blitz rules.</p>

<p>I've seen all kind of forfeits because of this reason in blitz games, including one player taking one of his own pawns.</p>

<p>However, the most common case for illegal moves is neglecting that the King is in check or unprotecting the King. In this case where the King is checked, if the opponent takes the exposed king and presses the clock, are they making a new illegal move, hence voiding the previous one?</p>

<p>I've been told that taking the King, even as a kind of "joke" to expose a previous illegal move, is an illegal move by itself, because the King cannot be taken, only mated. Is this true under FIDE rules?</p>
 12: <p>This is actually very common in blitz games.  Taking the opponent's king is considered to be a claim of an illegal move.  You can see my answer to <a href="https://chess.stackexchange.com/questions/181/i-made-an-illegal-move-what-happens">this question</a> for some more information.</p>

<p>Since taking the opponent's king is a <em>claim</em> instead of a move, it immediately ends the game.  In blitz games, the player "capturing" the king wins.  In slower games, taking the king is in fact illegal, and if a player did this, the position would be reset to the position before the first illegal move, and then any touch move rules would apply.  If there were time pressure considerations, one or both players could have two minutes added to their clocks at the arbiter's discretion.  </p>

<p>From the <a href="http://www.fide.com/component/handbook/?id=125&amp;view=article" rel="noreferrer">FIDE Appendix for Blitz and Rapidplay</a></p>

<p>Article A.4.c:  </p>

<blockquote>
  <p>An illegal move is completed once the opponent’s clock has been
  started. The opponent is entitled to claim a win before he has made
  his own move. However, if the opponent cannot checkmate the player’s
  king by any possible series of legal moves, then the claimant is
  entitled to claim a draw before he has made his own move. Once the
  opponent has made his own move, an illegal move cannot be corrected
  unless mutually agreed without intervention of an arbiter.</p>
</blockquote>
 103: <p>Capturing the King is an illegal move according to FIDE.</p>

<p>Refer to Article 1.2 from the <a href="http://www.fide.com/component/handbook/?id=124&amp;view=article">Laws of Chess</a></p>

<blockquote>
  <p>The objective of each player is to place the opponent’s king ‘under
  attack’ in such a way that the opponent has no legal move. The player
  who achieves this goal is said to have ‘checkmated’ the opponent’s
  king and to have won the game. Leaving one’s own king under attack,
  exposing one’s own king to attack and also ’capturing’ the opponent’s
  king are not allowed. The opponent whose king has been checkmated has
  lost the game.</p>
</blockquote>
