"Influential"? No, probably not. He seems to have been a competent teacher (though apparently he was a terrible mumbler who was frequently hard to understand) and a competent scholar. Note: I'm not saying he was mediocre: You don't get a named professorship at Oxford by being ordinary!
People who should know have said that his essay "The Monsters and the Critics" is a major work of Beowulf scholarship and changed the way people looked at the poem. This may be the most influential thing he wrote as a scholar. The rest of his work tends to be noticed mainly because of who he became later with LotR.
There's a good very short bio by The Tolkien Society which comments "His academic life was otherwise largely unremarkable. In 1945 he changed his chair to the Merton Professorship of English Language and Literature, which he retained until his retirement in 1959. Apart from all the above, he taught undergraduates, and played an important but unexceptional part in academic politics and administration."
He does have one handicap: His approach to English (through philology) was on the way out even during his career and being replaced by a focus on modern literature. (Tolkien was involved in academic battles between the two schools). His work is consequently "out of style" and probably more likely to be cited today in language and medieval history departments than in English departments. See the first page of the article by Jill Fitzgerald (which, unfortunately is all that seems to be online) for more discussion.
In the end, I think it's fair to say that had he not written The Hobbit, LotR, etc., he'd be remembered as an ordinary mid-century Oxford professor, one of the many Oxfordians who were part of C. S. Lewis's Inklings club.
(Edited to add some detail developed in the comments.)
