Got a Beeman R7 springer with a Airgun scope on it. Going to go back to my receiver sight, but was wondering if my scope can be used on any of my Centerfire rifles, i.e. 308 Win?

As far as I know you should be fine, springers are supposed to be the hardest of firearms on scopes so it should be a step in the softer direction.

Back in around 1960 I had Thomas Bland, a London gunmaker, fit a scope to a Webley MK111 air rifle.
As they had broken a lot of so-called air rifle scopes on rifles like mine they put on one they used on their .375 magnum rifles.
It was still working when I sold the rifle in 1999.

I don't know what certain makers supposedly do to make scopes resistant to airgun 'recoil', but imagine it is just the same old image-movement system beefed up in the erector-tube hingeing and spring(s). Therefore, it might be better than many others if put on your centrefire rifle.

Very few scope makers had moved to that system by 1960, so Thomas Bland probably used one of the then-conventional reticle-movement models. H&H favoured Zeiss/Hensoldt and B. Nickel scopes but others like the Pecar, Kahles and Leupold offerings of that time were also tougher than anything made now.

Thought I may try it out on my 405 Grenadier that Tom is building for me.