From the Super Mario Wiki
Super Mario series
Super Mario Bros. 2
In Super Mario Bros. 2, they are the main modes of transportation of Subcon, similar to that of Warp Pipes from Mushroom World and the Real World; if a character goes down a vase, they will appear in a small room sometimes containing a key being protected by a Phanto. In some levels throwing a Potion on to a vase will allow the player to warp through the Worlds, acting as the Warp Zones of Super Mario Bros. 2. Potions won't work inside vases though. Cobrats sometimes lurk in the opening of some vases. Certain vases generate endless streams of enemies like Bob-ombs or red Shy Guys unless their entrances are blocked. Vases made a small appearance in the Super Smash Bros. Melee course Mushroom Kingdom II, which also took place in Subcon. In Super Mario Advance, some Vases also contain Balls.
BS Super Mario USA
Vases also appear in the pseudo-sequel, BS Super Mario USA, and can be used in the same way as Super Mario Bros. 2. In Subspace, Vases will lead Mario to Wart and they shall battle against each other.
Mario Kart series
Mario Kart 7
Vases appear as background items throughout the Shy Guy Bazaar course in Mario Kart 7. They maintain their original appearance and coloration from Super Mario Bros. 2. If driven into, they break and some items will fly out, namely Bananas. Some of them can also walk and move around, and they reveal many snakes when broken.
Super Mario World 2: Yoshi's Island / Yoshi's Island: Super Mario Advance 3
Vases of a much different kind later appear in Super Mario World 2: Yoshi's Island, here called Pots. These pots are about Yoshi's size, and are often found in castles and reveal contents (usually a key) upon breaking after a fall. One of these pots is even "powered up" by Kamek, forming Roger the Potted Ghost. Pots also make a cameo on Chomps' platforms in Super Mario Advance 3: Yoshi's Island, though they are not there in the original game.
Yoshi's Island DS
Vases make a minor appearance as a part of Virtual Boy Wario Land's vase-based enemy.
Mimi once had a vase in the game Super Paper Mario but when Mario breaks the vase by accident, Mimi says he owes her 1,000,000 Rubees. Mario eventually gets them. An exact replica of Mimi's vase can be seen in the house of Heronicus in Flipside in Super Paper Mario.