Can I burn out my multimeter's µA range by using a few mA of current?
Multimeters use a precision shunt resistor to measure current indirectly (known R, measure V, calculate I). Typically a low current range (uA to low mA) will have a larger resistance shunt resistor to generate a larger voltage drop. 

As a result, this larger resistance is no longer negligible at higher currents and begins to cause a measurable voltage drop. Additionally, the power dissipation can begin to cause resistance change and eventually it will kill the resistor. Typically, a fast-blow fuse protects the shunt resistor from continuous over-current. 

If you're wondering if running 10mA through the meter while in uA mode will damage it, the answer is most likely "no". If you run 300+ mA through the meter, then you will probably be in danger of blowing the fuse. If you try to use the meter as a volt-meter and forget to switch the leads, and connect it to a supply capable of dumping a lot of instantaneous power, you might be able to blow the shunt resistor before the fuse blows.