A couple days ago, I learned (most of) the guitar parts for the Beatles' version of "Twist and Shout." The opening guitar phrase reminded me of "Let's Dance On," so I learned that guitar part in order to compare the two. Both start with a four-note diatonic ascent (A B C# D in "Twist and Shout;" G A B C in "Let's Dance On"), then there's a considerable drop (a fifth in "Twist and Shout," down to G; a sixth in "Let's Dance On," to E), and then a bit of an ascent (up a second in "Twist and Shout" to A; two notes (F and then G) for a total interval of a third in "Let's Dance On"). Both songs also have rather lengthy chromatic phrases. There's a six-note string (A to D) at the very end of "Twist and Shout" and an eight-note string (F to C) in the second and third phrases of the introductory figure in "Let's Dance On."
I checkt the credits on The Monkees album, and "Let's Dance On" was written by Boyce and Hart, so 1) there's precedence for that Beatle influence (I remember something about their writing "Last Train to Clarksville" based on "Paperback Writer," although that's a couple years after "Twist and Shout" in the Beatles' catalogue) and 2) it's likely that the song was written specifically for the Monkees and - because they were modeled on the Beatles - it's possible that the resemblance to "Twist and Shout" was intentional.