Does internet use minutes on iPhone?

No they don't, "minutes" are for your phone calls and since you aren't calling someone when your using the internet (1) on your iphone it does not use minutes.

What it does use is your Wifi's bandwith (if you are connected to one) or your 3g signal (if you have the plan for it).

(1)=If you have a call running in the background---green tab on top---then yes you are using minutes