Thanks a lot for the theme
I don't think the Google add or search could jeopardise anything as far as being well indexed with other search engines.
I don't think no one in this business would dare to do such filtering, but who knows, as it's very doable (bots do not compile JS, but they can figure out an AdSense JS code is in a page).
Though, such filtering would be found out very fast by Google, and no AdSense challenger could afford not to be honestly listed in Google.
So I think such filtering just does not exists or has very little effects since for sure Google would fight back on such matter, and since they are first ... no one would have a real interest right now to risk being penalized in the Google search results.
Then, the add search can be very useful for any web site being well indexed, and even if, as phpBB, it has internal search functions, just because you'll offer better search results (I mean Google is better than any internal search function as long as your site is well indexed) while saving server resources (it's Google servers to work for your user here) and making some incomes.
The Google (add) search site is actually one of the goodies we are able to implement if our pages are well indexed, and this only could, IMHO, be one of the major reason to SEO a web site