For anyone not familiar with TinyURL, in layman terms it's a tool where users can enter long displaying URLs to get a shortened version. TinyURLs are often used where long URLs might wrap and therefore break, such as in email or social media web applications like Twitter. In more technical terms, TinyURLs are short, dynamically created URLs that redirect users to another intended URL via 301 redirect. Because TinyURLs "301" or permanently redirect, search engines should not index the TinyURL but instead should index and pass PageRank to the actual URL.
It is important to note, TinyURLs to paid links passing PageRank is a violation of Google Webmaster Guidelines and that sites like Twitter use nofollow techniques to prevent spam.
On their own, TinyURLs can be search engine friendly from a technical perspective. At the same time, I wouldn't suggest replacing your site's navigation with TinyURLs and would point out that tracking TinyURLs via analytics might be difficult.