I ran into this once when I was being relocated for a position with a new company. I had about 400 bottles or so that had to be moved from the LA area to the SF Bay area. I was first told by the move coordinator that it would be no problem, but then when the actual moving company showed up to do the pre-move walk through they said there was absolutely no way they would move alcohol of any sort. From a bit of digging I learned that this was actually a pretty common stipulation with most moving companies due to the liability. If they break a bottle or two there could be a lot of back and forth trying to determine an actual value. Plus, what if your 18 year old son signs for the stuff at the drop off point and they've just unknowingly given someone under 21 cases of booze.
At least those were the reasons they gave me.
After a lot of back and forth (this actually became a sticking point in the relocation as I had 2 weeks to figure out what to do), they eventually agreed to move it with the following caveats:
1) I had to pack all of the bottles up myself, and they all had to be clearly labeled "contains alcohol" on the outside of the box.
2) Those boxes were all then double-boxed into mover's boxes on the day of the move, again labeled on the outside.
3) At the dropoff point they would release the shipment only to me.
4) There was $0 coverage for anything broken in those boxes during the move.
So I would double check with any moving company you get, but if you're lucky like I was, having them move it is definitely way easier than trying to do it yourself. The only other alternative really is lots and lots of boxes (check your local BevMo, etc. for empties) and moving it yourself.
This is also a good time to have a "cellar cleaning" party or two and invite everyone you know over to drink through the bottles that aren't really worth moving.
I hope that's somewhat helpful!