I just moved in a house, and pretty new to all applicances inside.

I changed the old mercury thermostat to a programmable Honeywell one, because the old one can't stop furnace unless lasting a long time, say half an hour. One night the furnace continued burning until room temperature reached very hot degree even can't sleep.

The new Honeywell programmable thermostat seems good, usually works normally, but sometimes it seemed can't stop furnace neither. The room temperature will surpass 2-4 degree than the set one. For example, the set temperature 22 degree, the heated room tem. 25 degree. If I turn off heat at thermostat, it will always stop furnace after around half an hour. Is it normal?

I am really messy about these. Hope for help.

The thermostat is wired and set correctly should turn the furnace off and on at the prescribed settings. In some cases there are electronic adjustments you can make in the programing of the thermostat. Read your directions to find out.

If this is doing the same thing with the new thermostat that the old thermostat was doing it could be some other problem with the furnace.