helicopter question

The starting sequence will depend on the type of engine - number of engines - model of ship. But there's generally no problem with restarting an engine quickly. What does your story need? When you say 'turned off' - do you mean IN FLIGHT? Or say, after a landing? There is a shut down procedure - again depending on aircraft - that takes a bit of time.

We would need more context to give you a clear answer.

What era? Contemporary, some past era?

What type of helicopter?

Why is the engine OFF?

How quickly do you need it to be up and running again?

A Robinson R44 is one of the most inexpensive and ubiquitous private helicopters out there. It is also about the only modern, still in production, helicopters that uses a reciprocating engine (like in your car). Most all other helicopters use turbine engines.

Shutting off the engine can happen in seconds. Close the throttle and it's done. But, that might cause damage. For turbine engines, there is a mandatory cool-down period for the engine to be run at idle. It's usually two minutes. For the reciprocating engine, there is probably normally a cool down period too, but I'm not familiar with that.

Even if the engine is shut down, the rotors will continue to spin down for another minute or two.

Start up is more prolonged because of the preflight checks and sequences. It can happen really quick if the pilot is good and is taking shortcuts, but that always runs the risk of missing something important.

So you need for the 'rescue' to take place faster than the pilot had anticipated? This indicates that they are unaware of conditions on the ground? You want the pilot to set the ship down, thinking the 'rescue party' - will be on the ground long enough to warrant shutting down, conserving fuel?

Then - all of a sudden, out of the blue - the people they came to rescue - somehow show up and hop on board, - pursued by 'the bad guys/aliens/monsters'?

"Quick take off before they get here/they find us/ the bomb blows !"

"Oh, crap, I just shut everything down, it will be XX minutes, before we can take off!"

That's what you're looking for?

It's doable in that time frame and it shouldn't cause any issues. Just do the flippy switch things and make it sound plausible.

There could be longer term damage from doing such a thing, and it could cause issues with the engine at some point if it gets too hot during the start phase, but that's really not an issue in an emergency. The equipment will take a lot of abuse over the short term and keep working just fine.