The DS3231 datasheet says,
Avoid running signal traces under the package, unless
a ground plane is placed between the package and the
signal line. All N.C. (no connect) pins must be connected
to ground.
However, I know that it operates without them connected, because I breadboarded a prototype that way. I think I will go back and correct that when I get my better soldering iron back.
Considering that, it's likely that those pins are connected to some kind of internal local ground plane, or more simply that the unused pins are in close enough proximity to the actual crystal connections, that there could be interference coupled into the circuit. Grounding them would effectively eliminate such interference.
It's worth considering the part about adjacent lines as well. Low power crystal oscillators are very sensitive circuits.
Having said that, it doesn't seem to have anything to do with the original question about power drain.