What happens if a player is forced to draw from an empty library and has Teferi's Puzzle Box in play?

Drawing a card from an empty library does not cause you to draw zero cards — it causes you to attempt to draw and fail. Under CR 104.3c, a player who is required to draw from an empty library loses the game the next time a player would receive priority.

Teferi's Puzzle Box has a triggered ability that, at the beginning of each player's draw step, has that player put their hand on the bottom of their library and draw that many cards. This replaces the normal hand refill, but it does not protect you from drawing from an empty library. If you must draw more cards than are in your library (or any cards from an empty library), the loss condition triggers immediately.

The two effects are entirely independent. The Puzzle Box's ability requires drawing cards just like any other draw effect. If your library is empty when you are required to draw — whether from the Puzzle Box trigger, a normal draw step, or another effect — you will still lose the game under CR 104.3c.

Example: It's your draw step. Teferi's Puzzle Box triggers; you put your 2-card hand on the bottom of your library (now 2 cards), and you must draw 2 cards. You draw both, leaving your library empty. Next turn, the Puzzle Box triggers again: you put your hand on the bottom (say 3 cards), giving you a library of 3, and draw 3. Library is empty again. On the following turn, if your hand is empty and the library is empty, you must draw 0... but if any forced draw of even 1 card occurs with an empty library, you lose immediately.

In summary: Teferi's Puzzle Box provides no protection against losing from drawing with an empty library. CR 104.3c and CR 120.3 make clear that attempting to draw from an empty library results in a game loss at the next priority window.