because you cut it silly. fingers are not made to be cut. get it sewn up, or if you are a tough guy make it bleed really bad then wrap duct tape around it. seriously.
You need to go to a doctor ASAP!! Since you posted this 6 hours ago I am assuming that it is night time and most urgent cares are closed so I would suggest going to the emergency room. If it needs stitches or glued it can only be done within 24 hours of the incident. It burns because there are ALOT of nerves and blood vessels in your hands and fingers which will cause it to bleed alot and hurt bad!! The best thing I can suggest is to make sure you wash it very well with peroxide or betadine, put a clean (perferably sterile) dressing on it and go to the hospital. If it is still bleeding you need to apply constant pressure to your finger with a clean cloth or gauze for at least 30 minutes. If the cloth or gauze becomes soaked with blood do not take that off just put another one on top of it. ALso keep your hand elevated about the level of your heart. and once again GO TO THE HOSPITAL. you may need antibiotics to prevent it from getting infected as well as a tetnus shot. Take care and hope this helped!!
GO TO HOSPITAL! You may need a Tetanus shot! One thing i am certain off is untreated cut and infections. I got in my life time over 80 stitches. Some i thought i could treat, but i was very wrong.
How deep is the cut? Is it still bleeding? How did you bandage it? What did you clean it with? did you use antibiotic ointment after you cleaned it? Was the bandage clean?
If its deep, its going to burn. A kinda stinging burning sensation, right? you could either go to hospital, or at least head to a pharmacy and ask whether the cut requires stitches. If its an old blade, or rusted in any way, you definitely need to get a Tetanus shot.
The worse thing you can do, is do nothing.