What happens (I run a stick..)?
The drive-shaft "drives" the tail-shaft and important parts that pump the lube are not turning, thus stuff gets hot, runs dry, and siezes up.
Most of the GOOD older pre-1980s manual transmissions could be towed indefinitely in neutral.
Many of the later ones cannot, including transfer-cases.
If you are towing more than a mile or so, and do not know for certain that it is okay for your manual transmission, better dis-connect the drive-shaft, secure the U-joint cups, and grass-string it up out of the way.
I have known of those with manuals that tow them a lot to install a re-circulating pump that draws the lubricant from the drain-hole, through a filter, a cooler, and then squirting it back into the gear-box, with the in-coming nozzle aimed at whichever bearing needs it the most, thus negating the time-consuming requirement of dis-connecting the shaft.