Diode and continuity mode are pretty similar. With one of may meters it is actually just one mode - get the beep if the diode measures less than some 50 mV or so.
Usually the test current is on the order of 1 mA. It can be a fixed current or just a resistor to the supply. The details depend on the meter.
The diode mode displays the voltage (usually mV) with the given testcurrent.
The continuity mode beeps below a certain voltage - what is shown could be the voltage as well, or the resistance or some arbitrary scale. This could be the same, as 1 mV is what one gets from 1 ohm and 1 mA.
Many chips are OK with 1 mA in either direction. There may be a few exceptions, but not many. The larger danger is likely from ESD.