Can a player be forced to adjust a piece? If one of my pieces is knocked off the board unintentionally, do I need to place it back on the board?

Can I continue to play with one or more pieces not on the board (but still have it in effect - for example a rook that was on A1 but has been knocked off the board still "watches" the vertical and horizontal)?

Can I place it back on the board only when I intend to move it?
When FIDE tournament rules apply: If you displace one or more pieces – knocking off the board, intentionally or unintentionally, is a displacement – you have to re-establish the correct position on your own time (Art. 7.4.1). While it is not specifically mentioned, I would say that you have to do this at once; if you wait, that could be regarded as an attempt to "distract or annoy the opponent" (Art. 11.5).

In a regular game, if you proceed while the incorrect position is on the board, any move will be regarded as irregular, the correct position before the knocking off will be reinstated, the game will continue from that position, the arbiter will set the clock and give penalties. In rapid and blitz games, after the opponent made a move, the game will continue without the "vanished" piece.

USCF might have additional rules.