It could be a control board fault, causing the central heating pump to run continuously.
If this is the fault, you will be paying for the electricity running the pump.
I would recommend switching off the electric supply to the boiler when not required for long periods, to save money and complain a lot more to the letting agent.
Tell them that the fault could be affecting the safety of the boiler and that they will be held liable should any incident follow. Put this in writing to them (so that they cannot deny receiving notification of the fault).