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This is just a timer based on hours. Does the thermostat have an "i" button?
the filter blinking light will have nothing to do with the furnace running. The filter light is just a timer that goes off avery so may hourse as a reminder. It does not know if filter is good or not.
It is like setting a timer on oven...even if oven is off the timer works.
You can reset filter blinking by holding i button and pressing until filter stops blink.
As far as the furnace not heating that is a totally unrelated issue. For that we would need make and model of furnace
but do you understand this filter light has nothing to do with the furnace running?
Remove front cover on thermostat and give me model on thermostat
See a number like TH6350 or rth5220 or similar?