There are numerous issues which can arise from sharing an IP address. Many of these issues can affect SEO. Some examples:
1. If any website which shares your IP address sends spam e-mail, then your IP can be blocked by various companies. Later when you try to send site-related e-mail to a customer your e-mail will then be blocked. The issue can take weeks to resolve one the root cause is found. This issue can impact SEO since many sites use e-mail to confirm registration, send advertisements or otherwise promote their site.
2. If you share an IP address with a "bad" site, your site's IP can be blocked or served different content.
3. Your site can be blocked by various companies due to sharing an IP with a bad site. For example, many sites have various spam protection where the admin presses a single button to ban a user along with his IP.
I don't believe a search engine would take action against your IP unless they felt most of the sites on that IP were bad. There is precedent for such action but it would need to be quite bad to reach that point.
In short, if you are even remotely serious about providing a quality website then a unique IP should be considered. It is required for SSL, and usually costs only a couple dollars/month otherwise.