In chapter 23, did Theo win the chess game?
Yes, Theo wins the game played with an unseen opponent in Chapter 23.
Only Theo, bent over the chessboard, said nothing. White had moved the king's bishop since the last meeting. It was his move. (143)
Theo never sees his opponent, but the game nevertheless continues while all the heirs give their answers. Theo is somewhat surprised when the game comes to its conclusion.
Theo grinned down at the chess table. White had made another move, a careless move. He licked the cake crumbs from his fingers, wiped his hand on a Westing Paper Tea Napkin and took his opponent's queen from the board. At least he had won the chess game. (148-9)
The chess game is symbolic of a few things in the novel. First, that Theo, though he doesn't win the "Westing Game", does win in the end by getting much better physically and having a good life. The unseen opponent (Sam Westing) plays the white pieces, showing that he's not the bad guy he's been made out to be. And the taking of the queen at the end of the game is a clue that Crow (Mr. Westing's ex-wife, and hence the "queen") is the clue to the mystery of the Westing Game.
Source: Raskin, Ellen, The Westing Game. New York: Avon Books, 1978.