I've recently deleted a lot of tags (www.example.com/tag/example) because they were either useless or ranking for keywords for which I have articles on (and I want to rank those).
I checked the sitemap.xml and the tags were also removed there. However I now get a 404 error on Google Search Console for all those removed tags.
Is this something I should worry about? Does it take care of itself or should I manually ask Google to remove the URLs as well?
If you removed the pages you should expect that they now return "404 Not Found" status. Google alerts you to 404 errors in case you removed pages accidentally. When you purposefully removed pages and Google reports "404 Not Found" crawl errors, there is nothing you need to do. That is expected behavior.
It is a common fallacy that your Google Search Console needs to be error free. That just isn't the case. Almost all websites have some crawl errors reported for them. Crawl errors don't always need to be fixed. When there is nothing wrong, crawl errors won't hurt the rest of your site in any way. See this post from Google's John Mueller who addresses the issue: HELP! MY SITE HAS 939 CRAWL ERRORS!!1
