How much current does a typical multimeter draw to measure resistance?
Multimeters nearly always inject a low current from a constant current generator into the resistor under test. This means that the voltage devloped across the resistor is directly proportional to the resistance being measured. On different ranges, the constant current injected may be bigger (in order to get better resolution on smaller values of resistance) and it may be smaller for the Mohm range.

Regarding the actual current drawn from the battery this entirely depends on the measurement system's efficiency which is entirely dependent on the circuit of the meter (which we don't have). For the Mohm range, 1 micro amp injected into the resistor-to-be-measured will produce 1 volt across 1 Mohm so this current is likely trivial compared to the current taken from the battery by the measurement system and LCD - if it's an analogue meter this may take more current than an LCD.