If someone wants to remove a custom character tag for an OC from the site entirely, how would they go about doing that? Would it be similar to getting put on a do-not-post list?

I'm no authority here, so the following are just common-sense ideas to consider; best not go making any major changes of that nature on my word alone.

the situations i can think of where a char. tag would be removed are:

• current tag name conflicts with information at the source or in the image itself (i.e. incorrect tagging) -> replace tag with new name, or perhaps with unknown_character if no name is indicated anywhere.
• current tag name is inadequate (e.g. ambiguous or too heavily abbreviated) -> replace tag with more accurate name
• character isn't prevalent enough to warrant a dedicated tag at all -- this one I'm not too sure about; i couldn't find any policy about what the threshold would be. maybe someone who's been around longer can comment.
• character has actually been renamed -> replacing the tag with the new name sounds reasonable, but i would suggest linking a citation in the wiki showing that the name-change is authentic (e.g. artist commentary, a specific post, etc.)
• the tag is a threat to your privacy (e.g. personally identifies you) -> I'd start by seeing if you can erase it from wherever it originated, then see if any of the above cases become applicable

e621 doesn't treat Trademark the same way as Copyright. The only case I know where a character has been granted DNP status likely involved a corporate request.

magnuseffect said:
e621 doesn't treat Trademark the same way as Copyright. The only case I know where a character has been granted DNP status likely involved a corporate request.

Only example of that that I know is paddington_bear. Usually, a character owner simply has to repeatedly submit takedown requests.

There is an "unofficial" characters to avoid wiki page, but as Furrin Gok mentioned, character owners must continually submit takedown requests. Save for Paddington Bear (whose copyright owners legally demanded that all explicit artwork of him be taken down), characters do not get DNP status.