Yes, there are ways to damage the meter.
The first is over current, this means that you always need a load in series that keeps the current lower than the max rated current on the meter.
It also means that you shouldn't measure current by using the meter to short line to neutral (which is common sense for most people, but it has happened)
For example, my meter will accept 10A, this means that I could not put a load downstream of approximately 2kW , but no more than that.
It's also very hazardous to do this kind of testing, it's not best to find current with a meter as wires would need to be spliced or a non standard electrical connection to the wall.
It's better to use a clamp current meter, or to use a power meter like a kill-a-watt meter (which also measures current) and plugs in series with outlets.