Is there a way to remove the mildew smell from under tile without damaging the tile?
If your renting, your only concern, I would think, is the smell.  If there is a smell it will disappear as soon as the water dries up.  When I fixed my mothers house the smell was supposedly there for years. It was caused by a slowly leaking toilet. When it was fixed, it took some weeks for the floor to completely dry, and the smell went away.  You may have other problems later like dry rot, and mold dust if you ever remove the tile.  SO don't.  If the smell continues then there is still a leak somewhere. Tell the landlord to repair it again.