Is there a way to exclude from search results all posts that have a particular tag?
Yes, although it must be a part of some other search. That is, you can't search just for "All posts that don't have this tag".
As part of your search, include
-[tag] in the search string. For example, if I wanted to search something on Meta Mathematics, but not posts with support, then I'd include
-[support] in my search string. This will then give me my search results without any results that are tagged as such (or in the case of answers, whose associated question is tagged as such).