Can I put a SIM that I'm using in a Nokia Lumia in a iPhone?

I'm using a Nokia Lumia 800 and have a pay monthly sim only plan. Will I be able to put this sim in an iPhone and use it as my personal phone, just as I used my Nokia?

You will assuming they're the same size sim. The Lumia uses a micro sim as does the iPhone up to iPhone 4. Later iPhones use a nano sim which is smaller

If the iPhone is either locked to the same carrier you are using the Nokia with now, or unlocked, then yes.