Unlike the original 1st Generation Pokemon Games, Pokemon FireRed actually has two different Game Corners. The first Game Corner can be found in Celadon City, much like it was back in the original 1st Generation Pokemon Games. Similar to those same games, the only game that can be played in this Game Corner is that of a slot machine. You will still need the Coin Case to play the slots, and the coins you earn from playing them can be used on various TM Prizes and Pokemon Prizes.
The second and brand new Game Corner of the Kanto Region can be found on Two Island. Unlike the Game Corner in Celadon City, you can play two different games (Pokemon Jump & Dodrio Berry Picking) at this Game Corner. Unfortunately, these are Multiplayer Games, requiring you to team with other players using the Wireless Adapter. Furthermore, you don't win coins with these games, only Trainer Card Stars by achieving certain high scores in them.