The motor will consume only as much current that it requires at that point in time.
That said - if the motor is under-sized for the load, the extra current available from the battery can lead to motor damage.
There appears to be some confusion regarding my statement about how using a larger-capacity battery can lead to motor damage if the motor is too small to properly drive the load.
The damage that occurs is strictly thermal and happens when the motor tries to drive the load but gets too hot while doing so. With small cells (a small battery), the cells deplete fairly quickly and the motor temperature spikes to some high value, then decays as the cells decay. The motor often survives the abuse.
When you have much larger capacity cells driving the motor that is too small to properly drive the load, the motor gets hotter and hotter until something fails.
Something that I discovered when I was a kid. <Grin>
Note that this is only a problem when driving a motor that is too small for the load.