What happens if I put aspirin on the root of my tooth?

Chemical burn. You will cause a severe chemical burn. Aspirin is an acid. Don't do it!
Not a good idea. Aspirin has salicylic acid when when placed on the gums or the root of the tooth well cause immediate numbness that last for a little and then stops but well damage the root or cause an ulcer on the gums, and on the long run if this is a habit, it can turn carcinogenic.
Nothing good. The Aspirin will not help any pain from the tooth by placing it on the root. Since it is an acid it can burn the gum tissue around the root and cause more pain because of the tissu burn.
You will further. Complicate the issue and additionally have to deal with the pain of an Aspirin tissue burn. Go see your dentist asap for appropriate treatment.