I configure the Text-finder plugin to search a specific filename.

If the file does not exist, the following message is written to the log:

Hudson Text Finder: File set 'post_build_status_marker.txt' is empty

and the build is marked as unstable (I have "Unstable if found" checked, otherwise I presume the build would be marked as fail).

I think this behavior is wrong. If there are no files to search, then the regular expression is simply not matched.

I would suggest the following possible solutions (in descending order of preference):
(1) change the behavior so that if no files match the file list, this is not an error
OR (2) provide a checkbox so that the user can say whether or not "no matching files" is an error
OR (3) at a minimum, update the documentation (and help text on the configure page) to specify the existing behavior