I am considering moving my site to Amazon EC2 instances. Would an elastic IP affect SEO?
The reason this won't negatively impact your SEO is because of something called DNS. Basically DNS resolves your Domain Name back to your IP address. When you're setting up accounts on various sites for Webmaster tools, they're only concerned with the domain name. When they attempt to crawl the site, they will perform a DNS lookup to find the IP. If the site has moved, then they would see that in DNS and go to the site's new location to check if the content has changed.
For most sites on the web, the servers are managed by companies where the actual owners of the site have no control over what their hosts do with their IP addresses. For most larger sites (Amazon, Microsoft, Google, Etc), they work from a pool of addresses, either through VPS pools, or through load balancing between servers around the world. Because of this a company like Google would be unwise to limit search results to any content based on the IP address alone.
In regard to SEO, in the past they looked at sites based on their IP address and IP ranges to see if you were cross linking between your own sites to gain search engine influence, but now they focus more on established social media.
Plenty of websites that rely on search traffic are hosted on Amazon. Using their elastic IP is no more risky than getting an IP address from any other hosting company.
Google doesn't usually target penalties to an IP address. Sometimes lots of sites with different authors (some good and some bad) share an IP address.
Generally anytime you get a new IP there is a risk that the IP has a bad reputation. A good indication (but not guarantee) is to check the email anti-spam services if the elastic IP is blacklisted somewhere.
You can check the health of the IP here: http://whatismyipaddress.com/blacklist-check
Keep in mind if you don't find the IP blacklisted it doesn't mean it is clean. It just means that the IP wasn't used for sending email spam.