I believe the last one was Madonna's version of "Don't Cry for Me, Argentina," which made it to #8 in 1997 when the Evita movie came out. I think the last one without a movie tie-in was One Night in Bangkok, as mentioned by Sam890 below, from 1985.
Of note, Ariana Grande's "7 Rings" was #1 for 8 weeks in 2019. The song starts with the following lyrics, to the melody of "My Favorite Things" from The Sound of Music:
Breakfast at Tiffany's and bottles of bubbles
Girls with tattoos who like getting in trouble
Lashes and diamonds, ATM machines
Buy myself all of my favorite things
Rodgers and Hammerstein are credited co-writers, and I believe that this was their first #1 hit since Perry Como's "Some Enchanted Evening" in 1949.