digital thermostat doesn't shut off gas-fired steam boiler?

Yeah, something isn't right. If you switch the mode to "off" and the boiler is still on than it is either wired incorrectly or the stat is bad. You may have to throw the digital stat away but you should buy a new digital stat. Get a good one like a Honeywell. You don't want to go back to a dinosaur stat. A digital stat is much more accurate and runs your boiler a bit more efficient. Even the old ones go bad, and most pollute the earth because they contain mercury. Anyway, here is how you find out if it is the stat. Turn the power to the boiler off. Take the cover off of the stat and you will have two wires R and W. Disconnect one of the wires, doesn't matter which one. Then go turn the power back on to the boiler. If it does not come on then you know the problem is in the stat. Get a new one and install it. Now let me also tell you to check the stat wire from the boiler to the stat. Make sure it is not shorting out anywhere. Just look for obvious places it could be pinched. If the stat wire seems good than get that new stat. If that fails, call your HVAC company, your landlord is wrong!

I agree with( william b) those digital thermostats are an over kill , they need to be programmed and are fine if you got kids playing with the thermostat, but all you want is control of your zone valve , take it off and put it away to give it back to your landlord and get yourself a plain thermostat with a range of 50-90 degrees and put it on yourself just a simple 2 wire job and that will give you back your control of settings. good luck

throw the digital thermostat in the trash and go back to a manual, one