In all the years of watching the Open, it is very rare indeed that the tournament does not finish on a Sunday. They always somehow manage to get around weather delays.
Of course years ago a tie meant an 18 hole playoff on the Monday, the most famous being Doug Sanders missing a short putt on 18 and then losing the playoff to Jack Nicklaus the following day.
John Daly won his Open in '95 after an abbreviated 5 hole playoff on Sunday against Constantino Rocca.