In the first and second season of Fringe, much hay is made of Olivia's stepfather, and his sending her a card on her birthday every year (even having Olivia finding one, unaddressed, slipped under her door in S1E06, complete with spooky music and a long gaze off into the middle distance).
Was this ever resolved in any way, or just shuffled off and forgotten once the multiverse geared up?
In the new timeline created when Peter was "erased" from existence, Olivia makes a comment that she killed her Stepfather. It's not said how she killed him. It's not said if it was in self-defense, or if she just feels that her actions caused his death.
This comment is made in Season 4 Episode21, "One Night In October".
While it doesn't finish the story arc about the birthday cards, it does acknowledge that there was a storyline involving her Stepfather.
Here's the relevant quotes from this transcript:
PROFESSOR MCCLENNAN: (soul searching) For as long as I can remember... I knew something was wrong with me. (Bolivia walks in) My father -- he knew it too. He saw the darkness. He -- he used a... heavy hand to try and fix me. The brutality of it, I mean, it was... (Bolivia lingers near the entrance)
OLIVIA: I al -- I also came from an abusive home. My -- my stepfather w --
Later:
BOLIVIA DUNHAM: (driving through the dark, rural woods) What you said to John about your stepfather-- you were trying to open him up, huh?
OLIVIA: Yes. It's also true.
BOLIVIA DUNHAM: So what happened to him?
OLIVIA: My stepfather? (matter-of-factly) I killed him. (Bolivia not expecting that reply)
We can only guess as to when this happened. It may have been when Olivia was a child and shot him (as she did in the original universe, but failed to finish him off), or it may have been as an adult during the alternate events that happened in the time of seasons 1-3, or any other time.
Combine this with the fact that Olivia overcame her fear of her Stepfather in the original timeline (see Jason Baker's answer), I believe we can safely assume that the writers were telling us that the story was, in fact, resolved. Her Stepfather had no more value to add to the storyline.
Note: I capitalized Stepfather because I'm using it as a character-name, instead as just a noun.
The current relationship between Olivia and her stepfather is never explored again, other than an occasional mention (season 2 episode 19, "Unearthed", springs to mind).
Olivia seems to overcome the fear of her stepfather in season 3 episode 19, "Lysergic Acid Diethlyamide" (the one where they're all cartoons). The episode has a kind of Inception feel to it, with Walter and Peter (and also William Bell; long story) entering Olivia's subconscious to find her lost consciousness. Everyone else they meet in Olivia's mind (except Olivia herself) is immediately hostile to them, especially Olivia's stepfather.
Various dialogue in the episode implies that the hostility is a result of Olivia's own fears:
Walter: As far as [Olivia]'s concerned, one minute she was fine, the next minute she was set adrift in her innermost thoughts. It's her fears that are causing the siege.
However, the climax of the episode has Olivia standing up to her subconscious fears (here represented by her stepfather at the head of a huge military force. As Bell says,
William Bell: I suspect I know what has happened here. I think my experiment was right after all. You should have been safe inside your mind... Except it's you, and you have never felt safe. You are your own worst enemy, Olivia. You took the opportunity to let your fears overwhelm you, but you just fought back. In the end, you are as strong as Walter and I always believed you were. And now you know it too.
This would seem to indicate that Olivia has conquered her fear of her stepfather. There's no word on what actually happened to him following this, but Olivia doesn't really seem to care any more.
