can you still move the pieces? if so, just play as normal. if not, reply back
Can't move in online chess

Interesting. The original problem occurred ... and still persists on my iPad. I switched to a windows based PC, and the problem does not present.

I see it too and there's nothing wrong with my PC.
I recommend that you fill in a ticket, with all due haste:
http://support.chess.com/Tickets/Submit
Place yourself on vacation, if the time gets low.
p.s. Strange observation: When I use firefox, I see those strange yellow boxes, when I use Chrome, I don't.
p.p.s. Now I have them in my games. Maybe they've changed the interface a little.

I also cannot move any pieces anymore on my ipad. I like the site better than the app on the ipad but cannot use it for online chess now.

I've replicated the issue. Thanks for the report guys!
For the time being, use the iOS app to make moves on iPhone/iPad.

Just noticed a new change on the PC interface, using FireFox. Instead of the solid filled squares in yellow and blue for past and new moves, a bold yellow line is displayed surrounding the squares involved in the move. These appear the same as I initially reported on the iPad problem. The moves work on the PC, however.
It's my turn, the timer is running, but I cannot move. The opponent's most recent moves are highlight in yellow surrounding the source and destination squares, but are not filled in with color as normal. Any ideas?