Dave Holden is the blade runner testing new employees at Tyrell's on the premise that the escaped Reps might try to infiltrate there. He underestimates the Nexus 6 Replicant. It seems that the new replicants are difficult to detect even with the Voight-Kamp machine.
Unfortunately for Holden, he is also our first introduction to both the Voight-Kampff test and also what the Replicants will do to achieve their goal. The eye at the beginning is supposed to be Holden's (although it isn't actually Morgan's eye).
Holden gets shot. He gets his gun partially out, but is shot again. Leon leaves him for dead.
There were two hospital scenes with Holden and Deckard in the film, but these were cut out due to not adding anything to the story, (and perhaps detracting from it even). This scene can be viewed in the documentary On the Edge of Blade Runner, which is expected be on the Blade Runner Special Edition DVD. It is expected one or perhaps even both scenes will appear in a "deleted scenes" section on the new DVD.
In Blade Runner 2: The Edge of Human, Holden is kidnapped from his hospital by the human templant for Roy Batty (which is his name as well). Even though hospital surgeons claimed it to be impossible, Roy put an artificial heart and lungs into Holden, keeping him alive. Batty has been hired by the LAPD to find and kill the sixth replicant (an inconsistency in the film fixed by the Final Cut), and he believes that Holden was set up to be wounded by Leon, and Leon was assisted by the police in incapacitating Holden - how else could he have successfully smuggled a gun into Tyrell's headquarters? Batty goes on to claim that all blade runners are replicants, and shows Holden the replicant that was killed by a security system while breaking into Tyrell Corp - a female version of Holden. Batty says that Deckard is the sixth replicant. Holden is reluctant to believe all this, and says that he could be the templant for a line of replicants based on him.
They travel to Deckard's apartment, and Holden attacks Batty and handcuffs him to a pipe in Deckard's bathroom. Holden goes to watch Deckard and Sarah Tyrell (the templant for Rachael) at a dilapidated safe-house for blade runners, and makes a number of hypotheses about the conspiracy. He is forced by his failing artificial organs. to return to Deckard's apartment and confront Batty, and makes a correct guess that Batty has no idea whether Deckard is the actual sixth replicant and only had a hunch. They then wait for Deckard at the blade runner safe-house and attack him when he arrives. Batty and Deckard battle briefly and Deckard states that Batty must be the sixth replicant because he is so strong, and Batty claims that it does not matter whether either of them is a replicant, because Batty will just kill Deckard and get his reward. Hearing this, Holden shoots and kills Batty.
Later, Deckard and Rachael attempt to escape the burning Tyrell Corporation building, and Holden rescues them in his spinner. After they gain false identities and go live offworld, Holden (who is still unsure if he is human) goes to the shack Deckard hid in with the model of Rachael seen in the film - and finds her dead. Holden then goes to the police station to get better artificial organs. He correctly surmises that Deckard actually left with Sarah Tyrell (who wanted to replace Rachael as the one Deckard loved), and wonders whether Deckard did not know this, or did and just went along with it because he got what he wanted.
In Blade Runner 3: Replicant Night, Holden tries to sneak onto the set of a film Deckard is having a hand in which is based on his days as a Blade Runner in order to deliver a package to him, just as Deckard learns that the "person" playing Leon is actually a replicant. Holden is caught by security and brought to the scene being filmed - Holden's interview of Leon, and at its end the replicant shoots Holden with live ammo, killing him. The scene was all planned by a man named Marley, and the package Holden was trying to deliver was a persynth (living image of a person derived from records of their actions and personality) of the templant for Roy Batty.
In the Blade Runner video game, it is revealed that Leon stole Holden's badge and left it in the apartment the replicants were using. Fortunately, blade runner Ray McCoy found the badge when investigating the scene. Additionally, it is shown in the game that Holden is the third best shot in the Rep-Detect department, behind only Gaff and Crystal Steele.