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Your furnace vent should be double walled, class B vent pipe. It has an aluminum pipe inside, about a 3/8" air space and a galvanized metal cover. It is supposed to have a one inch clearance from any combustible material.
On your roof, looking at the top of the vent, there should be a 'cap' (they come in several styles, like an oriental's hat, 4 shaded vents, or just another layer with vertical splits around the pipe and a flat top), next there should be a 'rain ring' that fits the pipe very closely, and the 'pitched roof flashing'. The most likely place for a leak is at the rain ring since it just fits snugly around the pipe; silicone caulk around the top of the 'rain ring' should stop your leak. You might also look for exposed nail heads in the flashing and dab them with silicone too.
You are correct, the duct tape is unsafe in this application because it is combustible. I hope it works out well for you and you are welcome to any more questions.