Just like with visitors, search engine bots go wherever the domain and links in your site point to, and index links according to which belong to your site, independent of the IP address it's hosted on.
So changing IP addresses is just like doing a 301 redirect (without the redirect) - it won't affect your search engine results or position.
Two potential exceptions to this might be:
- If the new IP or its DNS servers are blacklisted:
Most major and experienced web hosting companies (but not all) will not give you an IP address that's been blacklisted. Also, depending on if the IP is only listed in SPAM databases, and not content databases, it won't affect your search engine rankings, but your emails may not get delivered properly.
To be on the safe-side, you can contact your web hosting company and ask them to check the IP for you. Also, you can check the IP yourself before switching over to it by checking various online database sites (some of which check multiple database sites at the same time).
- The new IP is in a different country or geographic location (e.g., The old IP is located in U.S. and new IP is located in Asia):
Reports seem to indicate that sites have encountered lower search engine rankings when their IP address was relocated to a different region. This might be more true of search engines like Bing, which places more of an emphasis on locality for ambiguous search results than Google does.
You can check the region where your IP is located by doing a WHOIS search for the owner of the IP address block by going to the registry located in your region, as listed here: Iana Registries by Region
Additionally, it may take a little longer to reach your website if your IP is located far from your target audience due to the physical distance, number of "hops", and network congestion that the data will have to travel through. You can do a
Traceroute to the new IP using online sites from different locations (e.g., Pingdom Tools) to get an idea if this might be a factor for you.
Lastly, it's wise to check Google Webmaster Tools to make sure you select your "Geographic Target" under: Configuration -> Settings