The lift key is the key to operate the lift in the underground base of Team Rocket in Celadon City. The key can be obtained upon defeating one of the Rockets inside the building. You will need to the key to go to Giovanni, the boss, and when you defeat him, you will obtain the silph scope.
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You don't
use the lift key to use the lift
You use it on the lift (1st Floor) to get access to Giovanni. This is your first battle with Giovanni.
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.
If you have completed the Gamecorner area(fought Giovanni and got the Silph Scope), then you'll be fine. The Lift Key's only purpose is so that you may use the Elevator there. Otherwise, you'll have to restart your game.