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You have a dead short somewhere in the control circuit if a fuse is blowing.Shorts can be troublesome to locate and may require popping a few fuses until you find it.Look first for any cuts, nicks or chew marks on control wires between units and thermostat.If all look fine, you will have to start removing components in the control circuit until we identify which is causing the fuses to blow.
Unless you can be on site for me to guide you, it is going to be easiest to call in a tech. Like I said, first look to see if a mouse or squirrel may have chewed a wire. If not We would start with taking the thermostat out of the circuit.Then the outdoor unit if you have ac,Then the gas valve, etc until we find which component stops the fuse from popping.
In otherwords, disconnecting each component one at a time