The Crest needs to be inserted into a slot on a statue of the Great KnightHegemol. Placing it there consumes the City Crest.
After walking through the opened gate, it closes and cannot be opened again. It is possible to open the ceiling above the gate by coming from the City of Tears and using the Desolate Dive/Descending DarkSpell on a breakable floor.
While the intended entrance to the City of Tears is to use the City Crest, there are additional ways to enter the City of Tears, such as dropping down from the Resting Grounds or coming from the Royal Waterways.