What kind of baseline criteria should we use to determine if a tag should be burninated? Is this something that happens on a more case-by-case basis? Is there a minimum number of tags that are needed before the tag can be considered for removal?
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It's almost entirely case by case.
If it's brand new, we in the New Tag Deletionist Cabal claim first blood. Bad tags, typoed tags, tags created just for single questions, etc, are all fair targets for our nefarious editing deeds. These are (mostly) done without any input from Meta, and are done by mere mortals instead of (just) moderators or SE staff.
Tags that contain multiple questions require a bit more thought and planning. First, what's to be done with the questions? Is the tag being replaced with another, effectively doing a merge? If so, how many questions are we talking about? Will people notice you doing all the editing yourself? Keep in mind that even retags cause a bump. When in doubt, it's going to be wise to consult Meta. This isn't just to make sure that the community is on board with the change, but also to solicit for help.
When consulting Meta, moderators and SE staff members may choose to help. They can either perform a quick burnination (almost always done by Jeff Atwood), or create synonyms, or perform merges. Most burnination requests don't end in moderator or SE staff performing the work, as of late. Usually either the community steps up and does it manually, or the request never gains support.
Simple burnination of a tag is sometimes not the right thing to do. For example, right now I'm soliciting opinion on cleaning up the find tag. Simply deleting the tag is a non-starter, yet something needs to be done.
Sometimes the tag itself, and all the questions inside, need to be cleaned up. Right now, there's an active effort to manually close and frequently delete all the questions inside career-development, as the entire tag and contents are off topic on SO and there's no site to migrate them to.
tl;dr Burnination is hard, let's go editing!
While there are a number of factors in tag burnination, the primary one is number of questions with that tag.
If the number of questions with that particular tag is less than, say, 1 to maybe 100 max, then burnination is unnecessary. Simply edit the questions to taste and remove the tag if you feel strongly about it. But do consult with the community first, of course, particularly if the tag has more than 50 questions. Bear in mind that editing a large number of questions will bump them to the front page (generally speaking), so that has to be considered as well.
If the number of questions with the tag is 100+ then editing is out of the question, both from a mindless labor standpoint, and from an edit bumping standpoint. Burnination is the only way. Open such requests on your friendly neighborhood meta!