Weird Al Yankovic’s I Want a New Duck has the same melody as Huey Lewis’s I Want a New Drug…
And Mary Had a Little Lamb (mi re do re mi mi mi, re re re, mi sol sol) is not the same melody as London Bridge is Falling Down (sol la sol fa mi fa sol, re mi fa, mi fa sol).
If you want to go with “really close”, then Jolly Old Saint Nicholas as almost the same melody as the Pachelbel Canon in D Major.
In live performance, the Mothers of Invention’s “Plastic People” was sung to the tune of “Louie, Louie.” For the studio recording, however, the song was given an elaborate arrangement that twisted the “Louie, Louie” melody beyond recognition.
Elvis’ “Love Me Tender”’ has the same melody as the Civil War song “Aura Lee.”
Brian Wilson is notorious for borrowing melodies. “Surfin’ U.S.A.” sounded so much like Chuck Berry’s “Sweet Little Sixteen,” that Berry sued for royalties and was awarded co-authorship of “Surfin’ U.S.A.” and “Surfer Girl” borrowed heavily from “When You Wish Upon A Star” from Disney’s “Pinocchio.”
Also, “My Sweet Lord” by George Harrison was involved in a suit alleging that Harrison stole the melody of “He’s So Fine” by the Chiffons. He was ordered to surrender the majority of royalties from “My Sweet Lord” and partial royalties from “All Things Must Pass,” the album which contained the single. Harrison, eventually had the last laugh, however. He ended up buying the rights to “He’s So Fine.”
John Lennon’s copied the melody for “Happy Christmas/War Is Over” from “Stewball,” an old folk song about a racehorse (you can find “Stewball” on Peter, Paul & Mary’s greatest hits album).
Numerous popular Irish folk songs use the same familiar melodies. The tune that many people know as “The Wearing of the Green” also goes with the song “The Rising of the Moon,” while “Young Roddy McCorley” has the same tune as “Sean South of Garryowen.”