In “Dark Crimes,” Jim Carrey wears a grayish beard, cropped hair, and the jacket and turtleneck of a silently dissident Slavic literature professor from the mid-’60s. His character, named Tadek, is actually a Polish police officer, and Carrey speaks in an accent that’s so whispery and vague I got the feeling that if he were playing a Polish plumber back in his “In Living Color” sketch-comedy days, he might have worked that much harder to make his inflections authentic.

Tadek is a detective who got demoted to the lowly job of records-room clerk after a botched investigation. Desperate to win back his status, and his dignity, too, he seizes on a murder case that no one else will touch, maybe because it involves a sex club frequented by his boss. Throughout “Dark Crimes,” Carrey broods and stares like an actor who’s out to muffle any hint of...