I have a hi-bird 250ST atv. The red wire from the ignition switch melted like a stick of dynamite's fuse while riding. Any suggestions for this problem?

 

This is why quads have fuses. Bypassing them is dangerous. The first thing I would do is put the in-line fuse back in. 10 amps is the value for the fuse. It should be located at the battery positive terminal and feed everything but the big fat starter motor cable going through the starter solenoid.

A melted wire (or for most people a blown fuse) means you have a short to ground. The wire that melted should have got uniformly hot over its entire length to the short. Thus inspect carefully all the wires (especially red wires) at every place where they come out of the wire harness. Look for additional places where there are melted wires. Melted wires means the current went there. That gives clues as to the path it took, and helps to find the short.

What are the wire colors at your ignition switch? I'm wondering if the red wire is the input side from the battery or the output side that feeds the lights and starter interlock circuitry. If the red wire is the input side of the ignition switch from the battery, and the wire on the other side of the switch isn't melted, then the short may be in the ignition switch itself.