One term that people sometimes use to describe movie houses that show short runs of older movies is "repertory theater." I remember encountering the term in the 1970s in Austin, Texas (where the Varsity Theater showed double features of classic, second-run, art-house, and foreign movies); in the 1980s in New York City (where the Regency and Thalia Theaters used same format); and in the 1980s and 1990s in Berkeley, California (where the UC Theater likewise adopted this approach).
The Web site SlashFilm offers this definition of repertory [movie] theaters:
Repertory theaters are theaters that primarily show classic movies the way they were meant to be seen. On the big screen. You’ve read about a bunch of them here on /Film. The Alamo Drafthouse in Austin, Texas. The New Beverly in Los Angeles, California. Film Forum in New York. These theaters, among others, build a reputation on showing fantastic old movies every single night.
As you can tell by the description, movie theaters like the Film Forum aim for significantly higher-brow audience than pays to two-thirds price to see six-month old Vin Diesel vehicles. But the older repertory theaters mixed in some fairly schlocky fare with the classics, and they (unlike the Film Forum) were quite cheap on a per-movie basis.
The complication is that repertory company is a standard live-theater term for a group of actors who put on a season of plays in the same venue; hence the venue is called a repertory theater.
In any case, "repertory [movie] theater" does have a somewhat tonier name than "dollar cinema" or "discount theater," and it does describe a certain kind of low-cost (in some cases), short-engagement movie house that usually offers relatively high-quality films from the past. A more general term for theaters that regularly show no-longer-new films is (as δοῦλος notes in another answer) "second-run theaters," defined at EnglishBaby.com as
(n.) theater where not-brand-new films are played
[Example:] I like going to second-run theaters because the movies are so much cheaper.