in general most multimeter voltage ranges use a 1 megaohm resistor between its positive and negative, in this case it generally takes about 1000 volt to permernantly damage most (not all) from overheating the resistor or just the voltage getting somewhere it shouldnt,
on the current range it uses a shunt, a trimmed piece of metal to give a voltage drop, usually in the ballpark of 0.1 ohm, this is what is rated for 10 amp, and is only used on the current range,
so to answer your question while in voltage range, the multimeter should not pull more than 1mA by design and you dont need to worry about the current rating,