Low voltage fuse blowing in furnace
Edit:

I concede that the answer to this problem depends on some specifics that are lacking.

If the OP fuse is a glass cartridge style, then it is likely that a slow blow (MDL type or similar) is required in place of the fast acting automotive type.  If the fuse is the automotive blade type as Mazura pointed out, then fast acting is the only option for those and the problem lies elsewhere.

Electrical contactors do draw relatively high current until the contacts seat into the closed position, so I also have to agree that a sticky A/C contactor could blow the fuse even if the contactor coil is actually good.  To test if the contactor is sticking, I would turn off the outdoor disconnect to the compressor and open the condenser access panel for inspection.   Turn the thermostat to "cool" and watch the contactor pull in.  If the contactor action is sluggish, that could cause the problem.  With power to the compressor turned off at the disconnect, this test could be performed repeatedly.