My thermostat went blank so I replaced it. Turned on the heat and nothing, checked the 5 amp fuse and it was blown. The fan turns on until I try to turn on the heat. I disconnected the low voltage going outside and the problem remains. I can turn on the heater by jumping the wires on the control board. Help please!!

Not sure where you are jumping? Since it doesn't do it on cool, wiring to compressor appears OK. The Heat relay may be shorted. Remove heat wire in air handler, try again, if it blows, it is more likely in thermostat wiring. Good Luck, keep us updated

I found a blown 1k ohm resistor. The heater works when I disconnect the blue wire, not trusting that though since it is the neutral.

Where are you located? Neutral for what? If for the main 120 VAC power, would not work with neutral disconnected and most likely not blue since
NEC code requires white or gray. In UK and most of Europe, blue is neutral.