The pieces are usually kept in the box and not even used that much. However, I took them out the other day to play and all the hex pieces were warped. How can I fix this?
A very nice set of laser-cut acrylic tiles costs about $100 ("How much do you love Settlers of Catan?")
To flatten the hexes themselves, stack them and bind them tightly with rubber bands in all 3 directions across the six sides.
To avoid the problem of the warped hexes entirely: get a sheet of clear acrylic that is big enough to cover the whole board and border. You could cut it to fit, cut it to fit and in halves for portability, or leave it square/rectangular. Place the sheet over the assembled board and play on the perfect resulting surface! It will flatten the hexes while you play, but I'd just keep a right-sized sheet with the game.