Even the title makes me think of it! (Let's do the time warp again!)
There were quite a few things while playing that makes me think this.

Bradley and Janet - Brad and Janet. Except in the game Bradley's not really a stereotypical nerd and Janet's kinda smart (but she is still naive in other ways) Also, their looks were and are very different from the movie's Brad and Janet. Haha.
"man that handles himself like a queen" or some such line - even though he isn't really a pervert Nathaniel (? the guy in the kimono not any of the other guys) seemed to be a tiny bit influenced by Frankenfurter.
Linds of course seemed the most influenced by Frankenfurter as he's a mad scientist pervert...the only difference between the both of them is that Linds I think never purposely killed anybody or cut out their brains.
Rafael - Like a thousand times more handsome than Riff Raff. I love
Richard O'Brien's voice but hate how he looks. (ahaha I am so shallow sometimes) Anyway, sorta a domestic but a domestic by choice! Awesome spin, that. As much as I adore Linds, Rafael is right up there too. Since I love them both I think they would make a good couple too.
I'm sure there's other things in there that kinda remind me of the movie but I know there was more than enough differences as well to make it not really be a parody of the movie. I'd be interested in seeing the characters from Date Warp do The Rocky Horror Picture Show. (along with some original characters added - they'd have to be anyway since there was no maid or tap dancer in Date Warp. Hahaha.)
I think all that was missing was a maid, a tap dancer, and a party scene. Well, if there wasn't a time limit of sorts. I understand why the rest was probably left out if they had even been considered.

And of course turned out that nobody was an alien from another planet. Drat.
I think somebody somewhere mentioned that they wished that there had been at least one possible yuri scene. I agree, the game as it is already is nice but what would've made things even better if there was at least one or two yuri scenes. I know that it would've been difficult to put such scenes in though since the only two other girls in the game are missing or in a coma or whatever. Here's where adding a maid and tapdancer, or even just the maid or just the tapdancer could've come in handy. Not only would it make for more TRHPS references/jokes/parody but then you've could've had some yuri scenes and maybe even full on yuri paths! But I know it must've taken a lot of work with the game as it was and is already so I can understand why some things were left out or maybe weren't even considered. Oh, and anybody else as bothered by me that sleeping with Linds led to the bad ending but the good ending never had a sleeping with Linds scene or mention of getting it on with Linds? Hahaha.