Barry Allen refers to a journalistic "Arkham exposé". Arkham, of course, refers to Arkham Asylum, the facility later featured in many Batman films. However, this episode aired before Arkham appeared or received mention in any of the Batman TV shows or films, thus the first TV or movie reference to Arkham was in a Flash TV show, not a Batman related adaptation.
Barry is walking Earl by a theater that is showing Batman (1989). Danny Elfman, who wrote the theme tune to this series, also wrote the score for that film and its first sequel Batman Returns (1992). In fact, when the Batman insignia is shown, the music briefly changes to something resembling the film's score.
Belasco's hideout is in the abandoned "Fillmore Central" theater. This is a reference to the iconic hippie-era venues, The Fillmore West and The Fillmore East.
While Barry is walking his dog Earl, he passes by a theater showing a double feature of Superman (1978) and Batman (1989). In DC Comics, the Flash, Superman and Batman are members of the Justice League. However, the implication here is that in the TV series, the other two are fictional characters, and only the Flash is real.