Using a multimeter to measure the current of a microprocessor
To measure current with a basic multimeter, you have to break the circuit and insert the ammeter in series.  That, is you must a series connection the 5V -> circuit board -> ammeter -> ground.  If you used the ammeter in parallel, which your description suggests, then low resistance ammeter would draw as much current as the power supply can provide.  This certainly could overheat the probe, although ammeters usually contain a fuse that should prevent this.  

It's also possible you accidentally shorted power to ground with the probe somehow, or if a second power supply is involved (quite likely for microprocessors), that you created a connection between the power supplies.

By contrast, voltage is measured in parallel, and voltmeters are high resistance so they do not (usually) interfere with a circuit's operation.

It is often easier to place a "shunt" resistor in series with your load and measure the voltage as an indirect way of measuring current, because you do with this without breaking a circuit on a PCB.