You find it on B4F in FireRed when you battle a Team Rocket Grunt in the far north corner he has a Level 21 Zubat and Koffing, if you defeat him, then he will drop the key. This will enable you to battle Giovanni. In Silph Co., you don't need a lift key to get anywhere, instead, you have to use the teleporters to get to the Card Key.
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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.
You use it on the lift (1st Floor) to get access to Giovanni. This is your first battle with Giovanni.
You get the Lift Key on B4F when you battle a Team Rocket Grunt in the far north corner. He even says "Who has the Lift Key" before the battle begins. He has a Level 21 Zubat and Koffing. Once you defeat him, then he will drop the key.
use the lift key to use the lift
To get the lift key, talk to one of the Rocket trainers you have already beaten. He will drop the key, then you can pick it up.