Yes, there is a rule that a player must make a move during her turn. If the player in turn has no legal moves and her king is not in check, then the game is drawn.
I realize that in a chess game, there is no reason to not move a piece.
Actually, almost all chess endgame theory, and thus a huge part of the whole game of chess is based on the fact that a player has to move a piece during her turn. I'll explain this by a simple example and some handwawing. Consider for example the following endgame:
[title "White to move and win"]
[fen "4k3/8/4K3/4P3/8/8/8/8 w - - 0 1"]
1. Kf6 Kf8 2. e6 Ke8 3. e7 Kd7 4. Kf7 Kd6 5. e8=Q
If, at her 4th move, Black didn't have to move the king, the game would be drawn, as there is no way for White to promote her pawn. However, because Black king has to step away, White gets a new queen and wins the game. (Note that this does not mean that a king and a pawn always win against a lone king; very often the game will be drawn, because also White has to make moves she would not want to make.) A situation like this, where having the turn actually makes your position significantly weaker, is called a zugzwang; for more examples, see for example the Wikipedia article
This and more complex endgames are the reason why one pawn advantage is often enough to win: You can exchange pieces until you get to an endgame where you can then often convert the extra pawn to a win. Very often the winning technique of king and pawn endgames are based on the fact that the winning side can force the losing side's king to step away from defending important squares simply because the king cannot stay on the best square forever. The winning technique of many other endgames is then based on the fact that the winning side can threaten to convert to a winning king and pawn endgame, restricting the options the losing side has.