Is there any item that makes you not need to rest after using a move that requires rest?

Question

Other than that one use berry.

Answer 1

You mean immobilization after Hyper Beam, etc? No, you cannot resist it

Answer 2

There isn't anything for moves that make you need to rest on the next turn.

But there is the Power Herb, which allows you to use these attacks right away, without the "charge up" turn:
Bounce, Dig, Dive, Fly, Razor Wind, Shadow Force, Solarbeam, Skull Bash, Sky Attack.
It is one-time use only.

Answer 3

For Solar Beam, there is a way to narrow the process down to one turn. If your Pokemon (that has the Hyper Beam move) uses this attack during (intense) sunlight (like as a result from Sunny Day, for instance), then it will not need to spend a turn charging. And no Power Herb required.

HOWEVER, there isn't a way to do this the other way around, like for Hyper Beam, Giga Impact, or any of those Move Tutor abilities given to the starter Pokemon. So, to be specific to your question, no.