You get it from a rocket from the rocket hideout in Celadon city.
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In the game corner you'll beat a man the the lift key will appear next to him. Hope this helps.
In Pokemon Yellow, the lift key is in the Rocket Hideout under the Celadon Game Corner. One of the Team Rocket Grunts will mention the Lift Key before you fight him. Once you defeat him, he will drop the Lift Key. He should be in the northwest room on the 4th floor down (B4F). To get to the Rocket hideout, you need to use the switch behind a poster.
It allows you to use the elevator in the Rocket Hideout, which allows you to reach Giovanni at the end of the hideout.
use the lift key to use the lift
You can find the Lift Key in the Team Rocket Hideout on Pokemon FireRed. It is dropped by a Grunt on basement floor 4 after defeating him.