The stove will keep running, but I imagine it won't hurt anything though. Although I would personally shut it down and clean the stove rather than let it sit with ash and coal in it.
Best bet is fully clean and protect the stove before you leave that way no rust will form and you will be ready for next season. Removing the stove pipe and cleaning it and closing it off from the stove is also the best bet. That way moisture from outside will not have it's way with the interior.
If you do search on summerize your stove there are many good tips, including putting a 60 watt light bulb in the stove for moisture control.
Have a safe trip.