Dave’s WordPress Live Search only works when logged in as admin

That plugin uses the AJAX API as it should…

http://frome.fm/wp-admin/admin-ajax.php?s=ho&action=dwls_search

… but when not logged in that request fails (from HttpFox)…

07:23:58.923    0.155   470 0   GET (Aborted)   NS_BINDING_ABORTED  http://frome.fm/wp-admin/admin-ajax.php?s=ho&action=dwls_search

Since you say that the plugin works when you are logged in the Ajax callback is likely hooked to wp_ajax_* but not wp_ajax_nopriv_* The latter is required for requests made by non-logged in users.

You will need to find the plugin code that hooks the AJAX callback and add a second line of code to hook to wp_ajax_nopriv_*.

It is possible that the plugin performs other checks, however, so that may not be adequate. It is impossible to say without digging through the code.