No, your tooth cannot bleed. When your tooth comes out, you see the blood inside the tooth. That blood is from your gum. It comes out into the tooth when it comes out.
Enjoy this answer!
Yes it can bleed from the soft tissue inside the tooth which is called the "Pulp". The dental pulp is a collection of nerves and blood vessels which can bleed if damaged. For instance if you fall over and accidentally break your front tooth exposing the dental pulp then you will see it bleed. Similarly if the dentist drills to remove very deep tooth decay and exposes the dental pulp in the process, then the dentist will usually see bleeding coming from the pulp, if it is healthy.