Believe it not, this not only actually exists (and I’ve used it), but @andrew and @taronlissimore are planning to offer their own “Community Preview Program” - which will cover more than just headphones, but include amps and possibly sources as well.
The other place you can do this is via “The Cable Co.” and their “Lending Library”. Something I used a couple, or three, years ago to audition the Abyss AB-1266 and HiFi-Man HE1000 … as no-one I knew had either of them at the time.
The process was smooth, and the non-refundable deposit can be applied to any actual purchase their (not just the headphones you chose to audition). But obviously it helps if you’re just trying to narrow down a definite purchase.
I think the community program that’s going to be offered here is going to particularly useful though, as it’ll help generate a broader array of impressions on the available gear.