There is no tag for English literature or Old English literature. If the former will not be added because it would fit most questions on this site, I'd propose the latter be added, as a tag limited in scope but also useful.
Old English in a linguistic sense (as opposed to "old English" with a lowercase 'o') is significantly different from later stages of English due to impact of the Norman Conquest in the 11th century. There is no similar watershed moment that separates Middle English (1066 – late 15th century) from Modern English. This is why Chaucer's Middle English and Shakespeare's Early Modern English are significantly easier to read than Old English: the difference with Old English is not simply due to a different distance in time but also to the enormous influence of the Norman French that the conquerors brought to the "sceptred isle" (apologies for the anachronism).
The issue of how to tag Old English literature was raised again in the chatroom in late July – early August 2021 without reference to this meta question, except to say that it wasn't very conclusive. As a result, we have started using anglo-saxon-literature, with old-english-literature as a synonym. Why not the other way around? This is to avoid users typing
english into the tags field and picking old-english-literature for any stage of English that strikes them as old (Shakespeare's English, possibly even 19th-century English); the issue is common enough on English Language & Usage, for example.
TLDR: The tag for Old English literature is currently anglo-saxon-literature.
I think an old-english tag would definitely be useful. I think a middle-english tag would also be useful, for works such as The Canterbury Tales. There is specific expertise involved in questions about both variants, so a tag would be useful.
However, Old English and Middle English aren't separate languages, they're versions of English. So in this case I would not follow the [language-X-literature] naming convention, which is used for questions that are about non-English works, so as to make this distinction clear. I.e. use old-english, not old-english-literature.