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My furnace is running almost non stop, there is heat but it is not blowing up through any of the vents
This means that the furnace isn't getting sufficient return air and would cut out on limit not pressure switch. This would be a 4 blink code not three. This is caused by a dirty filter which you said is new, a blocked evaporator coil which is the A/C portion that sits on top of the furnace. This is probably the problem because if this is blocked up it will restrict the amount of air you are getting through the vents. When you take the doors off it allows the furnace to pull more air to combat the restriction.
It could be there is not enough return or your exhaust vent is blocked try taking the exhaust pipe off of the top of the furnace and see if it works. Again do not leave it like this it is only for testing the vent function