The reason furnaces (as opposed to heat pumps) keep running is to cool them back off since the burner is pretty hot. It may be a safety issue (too much undistributed heat could make the furnace hot to the touch and possibly result in a fire), but I suspect it's also done to extend the life of the furnace from large temperature swings with the side benefit of a bit of extra heat distributed through the home rather than up the flue.
From a heat pump, I suspect the temperature inside the blower is below the setting used by a high temperature switch in a furnace (which is what keeps the blower running). Also, circulating the coolant in the lines is what causes the heat to be generated, so if you were to do this, at what point would you stop?
And finally, the heat that is in the vents will naturally rise, so turning off the blower will not result in that heat being lost. If your vents are in the ceiling, the rising hot air will go back into the vent over time even if you blew it out. And if your vents are in the floor, the heat will rise as colder air displaces it, without having to run the blower.
So can you get a little more heat out of the system? Perhaps. But I believe it will be very little. And since drafts in the home make you feel cooler in the winter, the result of saving that extra heat may result in you feeling the need to turn the thermostat up a bit more.
If you had a DMM or VOM and know how to use it, that would make diagnosis easier.
However, the approach to take is to divide and conquer. Identify what component(s) are causing the signal for heat. Is it the boiler controller, the thermostat, or a defect in the wiring?
I would begin by inspecting the areas where anything has changed since it last worked okay. Look for anything amiss: stray wires, debris, bird nests, junk.
Next, I would disconnect the thermostat from the wiring to the boiler. This should be just a couple screws to loosen and then pull the wires out from under the terminal. If they are not naturally stable, wrap each individually with tape—masking, electrical, scotch, duct: whatever you have handy. If the system continues to heat, the defect is probably1 not the thermostat.
At the end where the wires leading to the thermostat connect to the boiler controller, disconnect them, being sure to record which goes where—use a camera, labeling the wires, or writing on paper. If the system continues to call for heat, the fault must be in the boiler controller. Otherwise the wire itself has a flaw. If there are unused wires in the bundle, you could try swapping unused wires for the needed one.
If you get here, it is likely the new controller is either defective, or more likely, not correctly connected after its replacement. Double, triple, quadruple check the connections and everything else which could go wrong. Reread the installation instructions to see if maybe there is a configuration option which might relate to the problem.
1 There could be several problems at once, so repeating tests of all items might be needed after the first round of repairs.
Best Answer
The heating fan will run for a cool down after the heating elements are turned off. If you listed your Furnace Model and your thermostat model we could check the wiring for you and suggest what you need to do.
But yes it sounds like the Thermostat is wired incorrectly.