If you aren’t changing servers, and the hosting is the same, then an IP change shouldn’t affect your basic EE installs. I changed the office location of an inhouse server with an EE install last year, and changed ISPs, and had to reassign the server’s IP (using OS X Server). Didn’t have to monkey with EE at all. Now I’m not sure if all servers will react the same, but I think you should be in the clear.
I’ve got another EE install on an unmanaged VPS with a dedicated IP, (Debian on Plesk) and I can’t see how an IP change would affect it. I’ve also manage a dedicated install on RedHat, and I don’t think an IP change would affect it, either.
There may be some instances where a specific addon might need to be adjusted, say if you have an ssl cert assigned to a specific IP, or some other IP specific addon setting, or outside service reference. But overall, you should be ok.