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Jason was killed by the regulators in Adytum. Their motivation to do so was to have a reason to rid the Boneyard of the gangs, most namely, the Blades. One night, Jason was trying to sneak into town to see his father, but the Regulators killed him, and placed his head on a stake for Jon to see. Razor is in possession of a holodisk that shows the conversation between two regulators prior to Jason's death, and the truth about the situation.
Jason's death only serves as fuel for the conflict within the Boneyard. Believing the Blades are responsible for his son, Jon pleads the player to kill Razor, their leader.
Before he was cut from the game, there was a quest to rescue him from the Rippers, a gang also cut from the final version of the game. They kidnapped him, and were threatening to kill him.
Jason Zimmerman does not appear in any Fallout game. He was supposed to appear in Fallout, but was cut from the final version of the game, for unknown reasons. His dialogue file can still be found in master.dat, though it is unused.