What would happen if you use warm base on a compressor ice cream machine?
Firstly, it will work and you will get ice cream.  However, cooling the base (a) decreases the time it take to freeze, and (b) potentially improves the flavor of the base (as ingredients meld together).  Let's say you are not concerned with "b".  The more quickly you freeze ice cream, the smaller the ice crystals...the creamier the texture.  So, in any ice cream making scenario, it is better to begin with cold base.  How much better?  How much creamier?  It's probably a matter of preference in the end.  It sounds like you probably have an ice cream maker with a compressor.  Why not make a double batch of base, freeze one immediately, place the other in the fridge for 24 hours.  Freeze that one the next day.  Taste them at the same temp/consistency, and tell us here what you've learned.