This is currently not possible and such feature is not planned for Firebug.
However, the DOM panel will (almost certainly) be part of the native Firefox developer tools and searching for a string in the panel is valid requirement!
So, you can make a first step towards having the feature by reporting a bug in bugzilla.
https://bugzilla.mozilla.org/enter_bug.cgi?product=Firefox%20OS&component=Developer%20Tools