Hayley's quote "Is this one of those Twilight Phones, but only with horrible, ironic consequences?" is a reference to The Twilight Zone series, especially the episode "Long distance Call".
When Roger is lost in the city and trying to find his way home, he exclaims "Where's a trail of Reese's Pieces when you need one?" This is a reference to the film E.T. the Extra-Terrestrial, where E.T. follows a trail of Reese's Pieces.
Duper's line "Lucy, you got some 'sploding to do" is a variation of Ricky Ricardo's line "Lucy, you got some 'splaining to do" on I Love Lucy. Duper even mimics Ricardo's accent while saying it.
The woman in the picture Father Donovan points to while explaining that self-aware robots don't go to Heaven is Pris from the film Blade Runner.
Father Donovan mentions sign-language gorillas, a reference to Koko.
Just as Family Guy's Brian repeatedly asks, "Whose leg do you have to hump to get a dry martini around here?" Roger says "Who do you have to probe around here to get a Chardonnay?" Both are references to a line from the Broadway play The Boys in the Band, "Who do you have to fuck to get a drink around here?"
In the scene where Stan heads off to kill Roger at the museum, he personifies his gun by talking to it as does the titular character of the 1980s TV show Sledge Hammer!.
Stan's sarcastic "Thank you Larry David" to Francine is a reference to Seinfeld co-creator and writer Larry David, whose pithy and curt one liners were made famous on both that show and on David's Curb Your Enthusiasm.
When Roger and Gertie are visiting the White House and are in the Oval Office, Gertie tells Roger about all the good and important things that happened behind that desk and Roger says, "and under it too", referencing the Bill Clinton/Lewinsky scandal.