A WebmasterWorld thread has discussion about the importance of Sitelinks. Here are the key points:
- Sitelinks do not directly influence your search rankings
- Sitelinks can be removed but not changed via Webmaster Tools
- Sitelinks appear to be constructed based on your site architecture, as Google understands it
- Sitelinks can strongly influence your click-through rate on your search result listing
- Sitelinks, when removed, can be gone as long as a year or more
Those are some of the key points I pulled from that thread. Getting deeper into the ranking component. Although removing a Sitelink should not impact your search rankings, you need to understand that Google gives you your Sitelinks based on how they understand your site. So if you see a Sitelink that doesn't make sense, then maybe you need to think about your site navigation.
The WebmasterWorld talks in general terms, but this webmaster also posted a thread at Google Groups, which has specifics. In this case, the webmaster had "Fiber Optic Lights" as his Sitelink, but then Google changed it to "Optic Lights." The change happened around June 26th and he noticed a big drop in search traffic for "Fiber Optic Lights" but a spike in traffic for "Optic Lights." This clearly implies that Sitelinks are somewhat important, but yet - you need to understand that the Sitelink represents what Google understands about your site. Removing a Sitelink won't necessarily change the way Google sees your site.
In fact, in the Google Groups, Googler, Jonathan Simon said, "we don't take action on Sitelinks feedback for a specific site." He said that in the context of having a Sitelink changed, but it also may prove to be evidence of the above.
In summary, Sitelinks do not have a direct impact on rankings. They may increase your click-through rate on your search results. They do have a reverse affect, i.e. because Google feels you deserve a sitelink for a particular phrase, Google clearly thinks that page is relevant for that phrase, so you may rank well for that phrase - but that is the chicken before the egg scenario. :)