we have just moved into a house and have a backboiler ( the fire
doesn't work the landlord says it is due to a flue problem, I don't
think this can be right cos if it was then the central heating and hot
water would not work either) am I right in my thinking please?

Basically, you should have a copy of the Gas Safety Certificate. If there
was a problem with the flue, then the certificate would not be issued. He is
probably committing a criminal offence.

second, we have an overflow pipe that is dripping, The landlord put
this down to the power shower, put we have noticed that if we turn the
hot tap on the dripping stops for a while.

A leaking ballvalve in the loft tank. Using water reduces the level below
the overflow until the drip from the valve replenishes the tank up to the
overflow level again.

Christian.