Assuming it's running on a 9V battery like most multimeters are, and the setting is on MegaOhms.
How much current will be drawn from the internal battery?
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Technically, voltage cannot be drawn because its a potential energy between 2 points like gravitational pull. Thus, when the multimeter probes is placed on the resistor, it would draw the full voltage of the battery, the thing that is drawn would be current which causes the galvanometer on the multimeter to deflect.
*edit You could probably use another multimeter to measure the current draw by connecting it in series with the element you are measuring.
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.