If you cut off a finger, blood will flow. It'll probably flow a lot until you put a lot of pressure on it. I'm thinking about wrapping it tightly with thread, dental floss, or maybe even a rubber band, to stop the flow of the blood. Pellicle's suggestion to put a plastic bag over it is a great one - it'll block stuff from getting into the wound, and catch the blood.
Warfarin slows clotting. It's good for small cuts and scratches. It reduces the possibility of a clot forming on the heart valve -- the blood cells that may try to attach to the heart valve or the materials that connect it to your vessels may clot without warfarin - add warfarin, and these clots don't happen.
So - warfarin won't increase the blood flow for cuts where a lot of blood will flow.
But -- for things like impacts with other things, you can get some large, and often very colorful, bruises. (The blood escaping from damaged capillaries and vessels takes longer to clot, so more blood gets into the tissues. I dropped a marble block on my foot a few years ago, and, boy, was that attractive for a few weeks). If you get a hard hit to the head, there's increased bleeding risk - and it probably should be checked out.
OTOH -- if you take way too much warfarin, or you take or eat something that makes the anti-clotting effects of warfarin even worse (cranberry, grapefruit, and other things), you may wind up like a rat poisoned with warfarin and all the lovely critters who eat the rat - working its way up the food chain to birds, wolves, coyotes, and many other carnivores -- by having massive internal bleeding and, in extreme cases, even death.
This is the reason that we (speaking for myself and others who self-test weekly) test weekly, manage (or take advice from 'professionals' who know what the hell they're doing) - so that our INRs stay in range.
Personally, I don't worry about the big cuts - I'm pretty careful, just like the others here are. I don't want my INR below 2 for more than a week or so (preferably never, but low for a few days), but I'm more concerned about it being way too high.