No. The DNS change should propagate within a day or two and everything will be fine. Leave the site on the old server for a few days though, (couple of weeks is plenty), to make sure it can still be found on the non-propagated IP# during the interim.
We upgraded from a Shared Hosting to a VPS account on the same server. The company told us we had to delete the old domain from the server and then contact them so they could do some clean up on their end. Then, they set up our VPS account.
But no, it didn't hurt changing IP's at all. Just the downtime was nerve racking.