Removing/Hiding Content and User Tabs?

It sounds like you have the Google CSE Search integration module enabled. With this module enabled, Google CSE uses the "search" path provided by the core Search module (rather than creating its own "search/google" path) and also uses the permissions of the core Search module. The solution should be to disable the Google CSE Search module.

Assuming this resolves it I probably need to figure out a way to document this.

Is there any way to keep the CSE Search module enabled, but also hide/lose the tab for "Content" search?
It is rather confusing to our users to see both the "Google" and "Content" tabs. Disabling the CSE Search module removes the tie-in to the Google Analytics module for tracking site searches.

I am also looking for similar functionality as bgillard. No content tab, but the user tab can be there. Moreover, I would also like the search indexing to stop indexing (and increasing my DB size). Any suggestion would be helpful.

AFAIK you cannot enable the CSE search module and disable search indexing, unless you upgrade to Drupal 7. In the meantime, there should be some way to hide the tab whether thru permissions, hook_menu_alter, CSS, etc.

You can hide the tabs by using some CSS:

body.page-search div.tabs {
        display:none;
}

Tidying up the issue queue, and marking issues as won't fix where the version is no longer supported.