If you have been performing SEO since a long time, you may have witnessed issues with host server and IP address. The troubles with host server and IP address come into play when you frequently switch servers or use a common hosting podium. With common hosting podium you encounter issues like domain problems and host server is not proficient enough to survive traffic eruption etc.
IP changes should not affect SEO, but search engines might blacklist IPs that host sites that host spam links, thus if you host your website on the same IP, your content might not get indexed.
My suggestion is to simply try to stick with a set of IPs that only host websites you own (or trust).
Dario is pretty much right. The only other thing is if you build 10-15 sites to link to each other - having separate IP addresses makes them look more legit (whois needs to be private too). If it is obvious that you have sites pointing to each other that you control the value of those links is diminished. At least that's one theory...