It's hard to tell exactly what your problem is, however, here are a few thoughts---
a) It could be the thermostat, but, I'd bet against that. Most thermostats are "bulletproof", especially the Honeywell round ones. Try taking the thermostat off of the wall---disconnect it from the backplate----that will disengage the low voltage wires and if the furnace doesn't turn off, then you'll know that the thermostat is not the problem.
b) Try turning your power switch off---waiting 5 or 10 minutes and then turning it back on again. Sometimes, that is the way to reset the control module which controls the furnace.
c) In my (albeit, limited) experience, what "goes" on modern, natural gas furnaces are usually either the hot surface ignitor (certainly not your problem) or the control module.
d) Do you have an instrucional manual or booklet for the furnace? Check it and/or look up the manufacturer on the web and then, telephone them for assistance.