Hi Guys...
It's not so much that physical damage can be caused to the radios and or other sensitive systems during an engine start it's more a case of the microprocessors in these equipments not being able to handle the low voltage encountered by the current drain on the battery.
The radios require 14 or 28 volts (depending on aircraft type and equipment age) for correct operation and in the event that the aircraft battery is getting a little old or worn it will not be able to sustain a voltage high enough to power the starter motor plus the other equipment.
Inside the radios, there is a power supply unit which converts the incoming 14 or 28v to those voltage levels required for the various circuits, in the case of the microprocessor this unit normally requires 5vdc for correct operation. If the supply voltage drops too far, this 5v might well drop below the threshold level required by the microprocessor and it will freeze or stall, whilst other circuits in the radio that need perhaps 12v may not be affected to the same extent. Thus the radios built in fault detection circuits may come into operation and display a fault indication.
Generally speaking, simply recycling the radio should be enough to restore normal operation. The radio should be switched off and then back on again; the radio itself cannot do this automatically because the microprocessor has 'stuck' and is out of synch with the rest of the unit.
Basically if the radio control unit has digital, LED, or similar 7 segment frequency displays switch it to OFF before engine start.
Hope this helps...