What is the overlap in songs between the original 1967 version, and the Disney 2016 live-action version of the Jungle Book? (I.e which songs from the original appear in the new version?)
Until the film (or at least the soundtrack) are released on the 14th, it's impossible to know for certain. Amazon (and other sites) are offering presale on the soundtrack but they do not have a track listing yet.
So far, IMDb lists the following songs on their trivia "Soundtracks" page, which were common to the original film:
The Bare Necessities - two versions, probably one in-film version with Bill Murray as Baloo with a second pop version either as an additional release or from the end credits.
I Wan'na Be Like You - Performed by Christopher Walken
Trust In Me (The Python's Song) - Performed by Scarlett Johansson
From the Wikipedia page it seems that this may be a complete list. There is a section that discusses the music in the film and these are the only three songs listed as being common to the original film:
The musical score for The Jungle Book was composed by frequent Favreau collaborator John Debney. Favreau and Debney incorporated several songs from the 1967 animated film. "The Bare Necessities", originally written by Terry Gilkyson, will be included and sung by Murray and Sethi. "Trust in Me" and "I Wan'na Be Like You", written by Richard M. Sherman and Robert B. Sherman, will be performed by Johansson and Walken, respectively. Richard M. Sherman wrote new lyrics for Walken's version of "I Wan'na Be Like You". Some of the lyrics to "Trust in Me" were spoken by Kaa in the film's teaser trailer.