I have a fractured finger and my doctor used the method of taping it with the finger next to it with tape, can that cut the circulation of blood to it?

Probably not: The doctor performing the procedure should not have made the wrapping so tight as to cause impairment of circulation.

Yes if too tight: If there is swelling and it is too tight all you meed to do is loosen the tape.

Appropriate therapy: 'buddy taping' a broken finger to a normal finger is an acceptable form of treatment. Care must be taken not to apply tape too tightly, otherwise it is possible to cut off the blood supply. A simple evaluation of the fingers taped will tell you if there is a problem or not. If the finger(s) appear discolored, are painful or swollen, take the tape off and re-apply more loosely.

Yes: That method, "buddy taping" is standard. The idea is to immobilize the injured finger. If the tape is too tight, the circulation can be disrupted. Signs of this would be change in coloration, numbness or tingling. The tape should be loose enough to allow the fingers to move, but tight enough to prevent the fractured finger from bending.