Will a fan cool my Pi 3 without heatsink? I'm using it for some hard-core gaming and it's heating up enough that the thermal warning comes up and I don't want to use heatsink because the case I've got for it has a fan that fits on the inside and I don't want to move it to the outside(I have not used it yet) should I just sucks it up and move it to the outside of the case and buy some heatsink or should the fan do just fine?
Yes. Forced convection will improve cooling over natural convection. Especially if the latter is inhibited to a certain degree by the case. It is definitely worth a try even though it would of course work "better" with a heat sink due to the significantly increase surface area (after all that is the idea of a heat sink).
Yes, it makes a noticiable difference in temperature. Of course you would want a heat sink with it if your doing any overclocking, but you usually don't need one for regular use.
Yes, as mentioned in the other answers the flow you create will help the heat going away from your pi. But I must advice you to use a heatsink if you have it constantly over 70 degrees celcius. Not only for the pi to survive, but it can also be a fire hazard if you stuff the pi somewhere in a tv set or something.