Running on Empty (1988)

Warner (Lorimar Productions)

River Phoenix received his first and only nomination for playing the son of rebellious parents, Annie and Arthur Pope (played by Christine Lahti and Judd Hirsch, both previous Oscar nominees). The couple has been on the run for decades, ever since their radical days as students, when they set a building on fire and accidentally killed a man.
Their son Danny (Phoenix), about to graduate form high-school, is an aspiring musician, who’s been offered a scholarship to the prestigious Julliard, not to mention the fact that he’s dating a girl (Martha Plimpton), who is not aware of his folks’ past.
Though the moral dilemma at the center is intriguing—to keep Danny with them and destroy his career or let him go and never see him again—the plotting is schematic and includes improbable behavior. Fortunately, the good acting of the whole troupe and Sidney Lumet’s multi-nuanced direction elevates the feature above the level of TV Movie of the Week.
Oscar Alert
Oscar Nominations: 2
Screenplay (Original): Naomi Foner
Supporting Actor: River Phoenix
Oscar Awards: None
Oscar Context
The most nominated film in 1988, "Rain Man" received four Oscars out of its 8 nods, including Picture, Director, Actor, and Screenplay by Ronald Bass and Barry Morrow. 
The other Best Picture nominees represented a mixed bag in genre and quality: "Dangerous Liaisons" and "Mississippi Burning," each with 8 nominations, Mike Nichols's comedic fable "Working Girl," with 6, and Lawrence Kasdan's literary adaptation "The Accidental Tourist," with 4.
This was River Phoenix's first and only nomination in a category that included Kevin Kline, who won for "A Fish Called Wanda," vet Alec Guinness in "Little Dorrit," Martin Landau in "Rucker: the Man and His Dream," and Dean Stockwell in "Married to the Mob."