Here's what I did, and they don't look bad.
Used a towel to pat dry the tiles.
Used my fingers to press out the wrinkles and squish the expanding cardboard back together.
Laid them down between two towels, put a textbook on top, and pressed down to get some more water out.
Pressed out the wrinkles and squish cardboard together again.
Laid them down between two paper towels (paper towels are thinner and I wanted to make sure they stayed flat) and laid several textbooks on top.
After several hours, squished with fingers again.
Dry paper towels and textbooks again.
The next day, I pulled them out and laid them on a dry towel.
At this point they're holding together really well and are only marginally thicker than the undamaged pieces.
The damaged piece on the left is only a bit thicker than the dry one on the right.
boardgames.stackexchange.com/users/859/idbrii
Customer Service at Rio Grande Games will replace pieces for a fee upon request.
I ended up with a set second hand that was missing a tile, and they sent it to me.
I'm not sure how cost effective it would be if a lot of tiles are damaged though, compared to getting a new set.
boardgames.stackexchange.com/users/389/Rob Mosher
Manners dictate the clumsy culprit should offer to buy you a new set.
Beyond that, you can laminate the tiles.
That said, you have to see this from a cost-to-benefit perspective.
Unless this particular set is dear to you, I would buy another one.
Especially since laminating the damaged tiles also means laminating all of them; you don't want to make damaged tiles distinct from their intact brethren.
Short of that, you can try a one-sided pseudo-lamination by sticking plastic sheets only on one side.
boardgames.stackexchange.com/users/824/Mohamad
The sad answer is that the only real shot you have at saving wet die-cut cardboard counters is when they're still wet, as far as I know.
If you immediately put them on a tray, or in a plastic box (if in layers, separate the layers with wax paper), and put them in a very cold freezer and leave them there for a long time (i.e. weeks or months), then the water freezing will tend not to mis-shape the cardboard, and gradual sublimation will happen very slowly over time and your counters should dry out.
Taking them out of the freezer later will require care as condensation will occur on the counter surfaces: best to bring them into air that's as dry as possible and as cool as possible, and have lots of blotting material ready until the counters' temperature gets warm enough.
If all you're concerned about is functional tiles that don't have separating layers, you can try re-gluing with a decoupage solution (good decoupage glue will dry clear), but I have never tried that
and I suspect such a project would require great patience and care (and again, I suspect would meet with more success if done while the tiles were still wet/damp).
In terms of this particular game where it's important to have relatively non-distinguishable tiles, I suspect this solution isn't going to produce happy results.
In the long run, you may find that the easiest, least expensive (in terms of materials and time) solution is to simply buy another copy of the game (and ask the water-spiller to help defray the cost, if you feel comfortable doing that).
boardgames.stackexchange.com/users/862/Viktor Haag
I used an iron set to Cotten.
Placed on a granite counter.
Pressed back, then front.
I pressed hard and straight down leaving the iron in contact for a good five seconds.
Could do three to four at a time.
Wiped moisture off counter with paper towel as I went along.
Took a few cycles on the wettest tiles.
Left flat on counter over night.
The iron did not burn the paper and the paper did not stick to the iron.
Turned out great.
boardgames.stackexchange.com/users/6201/Dave weber
I had a go at the straight in the freezer trick and it worked a treat.
2 weeks of being frozen, they like they had just come or of the box.
I then put paper towel under and on top of them, before setting a mountain of cookbooks on top.
They did perfectly (Smallworld lives again!)
boardgames.stackexchange.com/users/9719/Cedwa38
I actually had the exact same issue - someone spilled water on my carcassonne tiles, and I blotted/placed it in between paper towels under a textbook.
When they dried, I put a bit of white glue where the cardboard was peeling.
When the glue dried, I ran the pieces through a laminator without the plastic a few times - they came out good as new!
boardgames.stackexchange.com/users/25310/Carcassonne
A more DIY answer: if the top has come off of the tile but not the bottom, then you could: Wait for the remainder of the tile to dry.
Cover in paper towel and press it with heavy books until it's dry and as flat as you can get it (two blocks of wood and a vice can help with this).
Print out the image of the tile (there are lots of images on the game's entry on BoardGameGeek: http://www.boardgamegeek.com/images/thing/822/carcassonne ).
Cut it out roughly, and paste to the top of your existing tile with PVA glue (white glue).
Make sure the edges of the image slightly overhang the edges of the tile, and that you cover the back of the image in only a thin but consistent layer of glue.
Take a bit of that transparent, self-adhesive book-covering plastic, and smooth it over the top of your pasted-down image.
Turn the tile over, and using a sharp knife (wallpaper knife, scalpel, that kind of thing) trace lightly around the edge of the tile, applying only a little pressure until you've cut through all of the overhang of the image and the plastic.
For bonus points, press it in the vice again to make sure it's as thin as you can get it; home lamination will nearly always be thicker than commercial printing, and waterlogged card will often swell.
boardgames.stackexchange.com/users/None/
Someone just spilled water on my Carcassonne tiles. The illustrations are coming apart from the cardboard (they're wrinkling).  What's the best thing to do? How can I dry them with best results?
Water damage on cardboard pieces
boardgames.stackexchange.com
<carcassonne><components>
boardgames.stackexchange.com/questions/2568
boardgames.stackexchange.com/users/859/idbrii
Lay them down between 2 paper towels and put textbooks on top. Try laminating the tiles once dry. You can also try putting them in the freezer and they will dry out over time without becoming mishapen. If all else fails, you can easily replace the tiles. You may find that the easiest, least expensive way is to simply buy another copy of the game or Customer Service  at Rio Grande Games will replace pieces for a fee upon request. Another option is to laminate the tiles.  One trick is to bot them between two paper towels and then lay several textbooks on top or to immediately put them on a tray and put them in a very cold freezer and leave them there for a long time, over time andyour counters should dry out. You can also print out the image of the tile at  http://www.boardgamegeek.com/images/thing/822/carcassonne.
Blot them between two paper  towels and then lay several textbooks on top.   Customer Service  at Rio Grande Games will replace pieces for a fee upon request.    You can laminate the tiles.   You may find that the easiest, least expensive way is to simply buy another copy of the game. If you immediately put them on a tray and put them in a very cold freezer and leave them there for a  long time, over time, your counters should dry out If you immediately put them on a tray and put them in a very cold freezer and leave them there for a  long time, over time, your counters should dry out. You can print out the image of the tile at  http://www.boardgamegeek.com/images/thing/822/carcassonne.