Toward the end of a game, I was offered a draw, however, the decline button did not work.  Since it was the other users turn I decided to let it go with out a response.  The user eventually ran out of time, however the game did not end and no user was declared the winner.  Eventually I decided to accept the draw and the game ended appropriately.

To decline a draw, you can simply make a move. However, since it was the opponents turn you could click, "Decline draw" I'm not certain on why it didn't let you draw. However if you go to your settings

http://www.chess.com/home/echess_settings.html

you can look to see if you have, "Auto-declare victory" unchecked. If that is unchecked than it will continue until your opponent moves.

Check that box and as soon as your opponent runs out of time it will declare a victory!