Throughout the Quarry Men/Beatles' existence, they played a total of at least 11 Jerry Lee Lewis songs in live shows (as dictated in Lewisohn, page 362-65), listed here chronologically.
The covering of Lewis' songs necessitated the addition of a keyboard player who could match (or at least try to imitate) his flamboyant and exuberant playing. Since none of the current members could do so, the responsibility fell to John "Duff" Lowe, who joined the Quarry Men briefly in early 1958. Lowe, however, resigned some time in mid-to-late 1958, and most of the band's Lewis covers date from a few years later. By then, Paul McCartney was playing piano well enough and could cover as needed - especially since he was still playing guitar at that point, as were Lennon and Harrison.
Of the 11 known Lewis songs covered by the Quarry Men/Beatles, official recordings exist of exactly zero. There are however, several bootlegs from the Get Back sessions of January 1969 that do show the band's affection for Lewis' work (despite rather unimpressive playing). These include "Whole Lotta Shakin' Goin' On", "High School Confidential", "You Win Again", "Great Balls of Fire", and "Fools Like Me".
And although never recorded by the Beatles, Paul McCartney performed "Mean Woman Blues" on MTV's Unplugged in 1991, but it was left off the resulting album.
Supposedly the Beatles recorded "Jambalaya (On the Bayou)" with Tony Sheridan in Hamburg, and it appears as such on some bootlegs.
However, I am very, very suspicious that these are actually the Beatles playing. I strongly suspect it's another band backing Sheridan.
Another Jerry Lee Lewis number from Hamburg - and this one definitely legit - is "When the Saints Go Marching In". Lewis released his recording of this traditional tune in 1958, and then recorded another version as part of the Million Dollar Quartet in December of the same year, however it wasn't commercially released until several decades later. The young Beatles were no doubt aware of the 1958 release, adding it to their stage repertoire from 1958-59, and later revived the number when backing Tony Sheridan in the recording studio in June of 1961, with the product occupying the B-side of their first commercial release.
The influence of Jerry Lee Lewis on the Beatles, then, is substantially less than Elvis Presley, Buddy Holly, or Chuck Berry (a pretty safe assumption based on the fact that they never did officially record any of his songs); but (much like Little Richard) no doubt Lewis contributed to the band's flamboyant and at times hysterical stage presence that they developed while in Hamburg.
Lewisohn, Mark. The Complete Beatles Chronicle. Pyramid Books, an imprint of Octopus Publishing Group Limited, London, UK, 2006.
This blog is a workshop for developing my analyses of The Beatles' music.