roof leaks may be difficult to diagnose.

if the whole roof is not leaking i wouldn't bother trying to cover the whole roof. lots of issues there - think wind, flapping, parachute..

it sounds like your leak is confined to the area around the chimney pipe. if you can safely go on the roof and look, you may find that the metal of the pipe or the flashing at the base is rusting away. a new one installed and sealed may be what you need - and not that expensive for a roofer to do.

if that's not it, the next cause may be due to old dried, cracked or missing sealant - the tar-like stuff - around the base of the pipe. if so, there's a product called Henry's that can be found in most hardware stores and that most roofers use to seal everything. it can even be applied in the rain. pretend you're icing a cake with black icing and spread it on. doesn't have to be neat - who's gonna see it - it just needs to cover the possible area of water entry.

to answer your question, typically - but not always - heat exhaust pipes are double-walled - so heat shouldn't be an issue. but i wouldn't chance it.

too, more rain will loosen the adhesive of the tape and no more seal...

hope this helps.

bsmith