Q:

Driving Mosfet, why is it burned? How should the circuit be modified?

I using MIC5018 to drive Mosfet (driving 48V，3A DC motor). Each power supply is powered independently. I am using a 48V, 3A DC motor. Before power-on, if it is connected to the computer (5V) and the signal is determined, it works normally. Once it is not connected to the computer before power-on, it may be burn the Mosfet.How should it be modified? 

A:

It's due to parasitic capacitance

Cgs gets charged trough Cgd. This capacitance may be found in datasheet

Basically, after turn on, you will have some small voltage on gate, that will partly open a transistor. It will have much bigger voltage on it and dissipating power. Add resistor (about 10k) between gate and source (ground) to negate that

A:

I suspect it would be enough to have a 5 V power supply which turns on at the same time as the higher voltage supply. (Note that neither the LM7805 nor the LM317 can handle 48 V. A stiff voltage divider would work for testing purposes.) Have you measured the MOSFET gate voltage when the computer isn't connected?
If the circuit has enough charge left (in the capacitor), the MOSFET could partially turn on. This could either increase heat dissipated, or worse, it will bring the MOSFET outside its safe operating region by causing localized heating on parts of the silicon. Vgs(th) drops with increasing temperature, and the regions that are hottest will start to conduct more, getting hotter and leading to localized thermal runaway.
Note: if my diagnosis is correct, you could also solve this by adding a pulldown resistor from the gate to ground so the gate voltage will stay low until the driver is powered and functioning.
For more explanation on what can happen when a MOSFET is conducting but limited by the gate charge, see the answers to my other question here.

