The Digistat, simply provides a 'switch' that makes and breaks a set of contacts, and that's all it does. The room thermostat should get its power from the programmer, so when the programmer is 'off' there shouldn't be any power going to the thermostat for it to switch, so I feel that the most likely scenario, as you suggest is that the programmer is faulty.
However, there are other things that could cause the boiler to fire if the programmer is 'off'. All are easy to check if you have a good understanding of electrical circuits, and know how to use a multi-meter safely. Only by carrying out electrical testing of the system, would you be able to determine exactly where the problem is.
Here are some things that could cause a boiler to fire, when the programmer is off.
1. Incorrect wiring of the heating controls. However, if this is a fault that has not always been there, and the wiring hasn't been interfered with you can discount this one.
2. The programmer has failed, so that the heating doesn't switch off.
3. If you have a motorised valve (or valves) these are what actually switch the boiler on and off when the central heating is required. It may have failed
4. If you have a frost thermostat that has been turned up or has failed, then this can run the boiler.
If the boiler is actually coming on and off under control of the thermostat (ie and not randomly) then 3 and 4 can be disregarded.
A simple test to check if it is the programmer is to remove the wire from terminal 4 of the LP522, insulate it for safety, and then put the programmer back on its mounting plate. Try to operate the heating, if it doesn't work now, then that would indicate that the programmer isn't switching off when it should.