Which wire exactly is burning up?? The way it works is the start button actually switches the ground for the starter solenoid, there is not power to the start button. The fuse is in the supply side which goes to the ignition switch and from there to the light switch and the rest of the controls.
You didn't mention, did the electric starter run at all? If not, i would guess you have either a bad solenoid which is on the back of the battery box, or you have a bad starter and its stuck. If you want to test the starter take the cable off the stud on the starter and then use a battery jumper or a set of jumper cables and try to spin the starter over by giving it power, do it with the ignition off so the engine doesn't start if it does spin over. Just hook one end of your jumper cables positive to the battery positive, it will be grounded through the frame so you only need the positive side for this test, and then touch the other end to the terminal on the starter.
If its popping the fuse though i would bet you have a bad solenoid. On the starter solenoid you will see two small wires and the two larger cables, one goes to the starter the other to the battery, the two small wires are power from the ignition switch and the wire that goes to the start button on the handlebars, when you push that it provides the ground and completes the circuit. The only way the wire could be getting hot and smoking and blowing the fuse is if the solenoids coil is shorted and its making it a dead short to ground when you push the start button.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....