What happens to a player in FFA when they attempt to draw from an empty deck?

They Burn Out. They draw as many as possible (none), recycle their trash into their Main Deck (randomized), choose an opponent to gain 1 point, then finish the draw instruction (Rule 431.2).

In FFA the only difference from 1v1 is that "choose an opponent" is a real choice among multiple opponents — you pick who gets the point. If the deck and trash are both empty, you'll Burn Out repeatedly, handing out 1 point each time until an opponent reaches the Victory Score (8 in FFA3/FFA4) and wins (Rule 431.3).

Note that points from Burn Out aren't subject to the final-point restriction that applies to Conquer, so a Burn Out point can win the game at match point (FAQ 7293).