From the comments it seems that some users do not know why and how the tag tag-removed is used on MO. So let me address this first. (At least briefly - some of the links below might add a bit more context.)
- Regular users sometimes use this tag for questions which are off-topic or outright spam.
- MO moderators sometimes use this to remove a tag from the system - by merging1 a tag into tag-removed. (Of course, they could merge into another tag - but situations where all questions in the removed tag would fit some of the existing tags are probably rare.)2
The usage by regular users probably isn't that important - since we're talking about posts which are eventually going to be deleted. So most occurrences of this tag on the site probably come from merging.
The usage by mods could be avoided in some cases. There is a process called burnination
which removes all occurrences of the tag without bumping. But this can only be done by SE staff - not by the local mods.3 (And SE staff is willing to do this only if the tag has already been reasonably cleaned-up. And only for situations which are too big for manual retagging - so basically only for tags with $\ge50$ questions.)
I do not know what is the stance of the moderators now - but looking at the past removals, at least at some point in the past they seemed to prefer merging over burnination.4
So to answer your question whether it would be possible to remove the tag-removed from MO, the best I can say is: Yes, this is technically possible. But, no I don't think it is likely to happen.
- In principle, the moderators could request from the Stack Exchange staff to burninate tag.
- But that would require at least some preliminary clean-up and manual retagging. If nothing else, at least retagging the questions which have only this tag. (At the moment, there are 35 such questions.) But maybe a bit more than that. (All would depend on the response of the SE staff.)
- Even if the CMs would be willing to burninate the tag, that would seem like a nonsensical move if the tag would be recreated again soon after that. (I.e., if the mods still want to have the tag tag-removed at their disposal and to use it as a target for merging. And if MO users want to keep this tag and use it for off-topic questions and for spam.)
Disclaimer: This is written from the viewpoint of somebody outside the MO community. I am in no way defending the existence of the tag tag-removed. And I am not advocating for the removal of the tag, either. (I will freely admit that I do not like this tag that much - at the same time I think that there are several problems with tags on MO which are worse than the existence of tag-removed.)
I merely tried to explain how and why it is (or was) being used on MO.
In any case, the tag tag-removed probably does not cause any actual harm. Still, if some question with this tag is bumped to the frontpage, it is a good idea to remove it manually. (Well, unless the question is very likely to be deleted anyway - for deleted questions the tags matter much less.)
Some additional links where you can find some explanations concerning the tag tag-removed:
1If a specific example can help clarify things, you can have a look at this question: Generalizations of "standard" calculus. At the moment it is tagged tag-removed. And in fact, if you look at the revision history, it shows that the original version had this tag. (Merging two tags replaces the tag in the revision history, too.) But if you check the Wayback Machine, you can see that it was tagged calculus in April 2013 and tag-removed in October 2013. So somewhere between these two dates the mods merged calculus into tag-removed. This is one of the tags discussed here: Tags that encourage off-topic questions.
2Probably the mods could use the tag untagged for this purpose instead of tag-removed. But basically the only difference would be that this tag cannot be used by regular users. Some information about the tag untagged can be found in the corresponding tag-info on Meta Stack Exchange.
3There is a feature request suggesting that: Moderators should be able to remove / burninate a tag themselves.
4I should probably mention the fact that MO was running on SE 1.0 for a long time. I do not know whether the option to burninate a tag existed before MathOverflow joined the Stack Exchange network. If that's the case, using an "artificial" tag like this was probably the best solution how to get rid of some tag without bumping any questions.