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 --
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.