Basically, if it's a stud game there'll be a single community card dealt face up on the river. The above post has a screenshot of it happening.
It's impossible to run out of cards in an Omaha game unless you're playing more than 11 players to a table. 11 players would require 44 down cards, 5 up cards and 3 burn cards for a total of all 52 cards in the deck.
In draw games, folded hands and / or discarded cards may be reshuffled and dealt again if the deck runs out (though there was some controversy over whether discarded cards should be used at this year's WSOP