On the rare occasion, a PGA Tour event faces a bad weather forecast, making it impractical or impossible to finish a tournament on Sunday. The PGA Tour does its best to make sure a tournament is completed with 72 holes of play, and sometimes that means going to a Monday finish.
However, you'll almost never see the PGA Tour go to a Tuesday finish, and you'll never see a PGA Tour event go beyond Tuesday. That's because PGA Tour regulations dictate only two circumstances in which a PGA Tour event could finish on a Tuesday (or beyond).
PGA Tour regulations dictate a tournament should finish on Monday if play spills over from Sunday to complete the event. However, when weather is so bad on Monday the entire field cannot finish, then the only way a tournament completes on Tuesday is if at least half the weekend field completes the 72-hole tournament on Monday. Then, and only then, will a PGA Tour event move to Tuesday. The last time a PGA Tour event had a Tuesday finish was the 2013 Tournament of Champions in Hawaii.
If at least half the weekend field cannot complete the 72-hole tournament on Monday, then the full final round is scrapped, including what's been completed. In that case, the winner is the 54-hole leader. In the event of a tie after 54 holes, the leaders go to a playoff as soon as is plausible to determine an official winner.
A PGA Tour event must go at least 54 holes for it to count as an official win. If a PGA Tour event doesn't go at least 54 holes, the tournament will still have an unofficial winner and will pay out the purse based on finish after the most recently completed round, but the win, money and points are considered unofficial in the eyes of the PGA Tour history books.
Conceptually, a PGA Tour event could be scratched altogether if it doesn't go 54 holes, and the event can be rescheduled and played all over again from the beginning.
The only exception to this rule is The Players Championship, which is the PGA Tour's crown jewel. The PGA Tour commissioner has the discretion to extend The Players Championship as long as he would like to complete a 72-hole tournament.