The Big Time Rush Mobile is even the same as the Monkeemobile. (without the customizations)
The Beatles Paul's bass can be seen in the background of the studio
Hell, Big Time Movie had Beatles everything.
They cover of "We Can Work it Out", "Help", "Revolution", "A Hard Day's Night", and "Can't Buy Me Love"
The beginning of the "We Can Work it Out" cover begins with BTR replicating the cover to Abbey Road, the beginning riff to "A Hard Day's Night", and then replicated scenes from the "Rain" and "Something" Music Videos.
The villain Maxwell Moon calls his plan a "Revolution"
The villain's henchman is Maxwell, with a Silver Hammer for a hand. They even say the phrase "Maxwell's Silver Hammer".
The spy girl they help is named Penny Lane.
And probably more.
Big Time Photo Shoot featured a famous pop star called Dak Zevon, who got famous because he took the role of a singing basketball vampire jock. Just guess what movies they are making fun of.
Possibly with Camille having a crush on Logan, as Erin Sanders' previous role on a Nick show dated a character named Logan.
On that note, Erin Sanders' friend dated a character named James throughout most of the last season, who, at least in the end, was pretty tight with the character named Logan. Probably not a Shout-Out, per se, but still an interesting coincidence.