There is a list here of many of the jabs at Fox. Specific Simpsons examples:
The Simpsons reached a disturbing new nadir in the episode "MoneyBART", its Couch Gag (storyboarded by subversive street artist Banksy) depicting the production of Simpsons episodes and merchandise taking place in a toxic sweat shop within a building shaped like the 20th Century FoxVanity Plate. This BBC report claims the sequence "led to delays, disputes over broadcast standards and a threatened walk out by the animation department."
In "The Springfield Files" Homer and Bart make plans to videotape an apparent alien visitor (and taking a swipe at Fox's "Alien Autopsy: Fact or Fiction" broadcast), giving us the second page quote.
"Sweet Seymour Skinner's Badass Song": The 100th episode's Couch Gag had the Fox logo bug appearing in the corner of the screen only for Homer to rip it off and the whole family stomps on it.
"HOMЯ" had Homer calling an automated stock price hotline, which worked by having the person calling saying the name of the company and the computer replying with the stock's respective value. When Homer asks "What is this crap?" the service replies "Fox Broadcasting: down 8 [points]," followed by him smiling.
"Thirty Minutes Over Tokyo" had Homer investing in "something called News Corp," followed by Lisa telling him that was Fox and Homer screaming "AHH! UNDO! UNDO!"
One of the earliest and more subtle ones may be in the Season 4 episode "Mr. Plow". When a TV commercial starring the family airs on late at night on an obscure cable channel, Homer exclaims, "It may be on a crappy network, but The Simpsons are on TV."
An even better example was in The Movie: During one of the scenes, a Commercial Popup crawler advertising the new Fox show Are You Smarter than a Celebrity? starts moving across the bottom of the screen, ending with "That's right, we even advertise shows during movies now."
Also there was Krusty bemoaning his good-for-nothing half-brother Luke Perry:
Lisa: But he's a big star! Krusty: Yeah... [with disgust] on Fox.
"Missionary: Impossible" begins with Betty White hosting a PBS pledge drive. At the end of the episode, she pops up again, this time hosting a pledge drive for Fox. She urges viewers not to let "crude, lowbrow programming disappear from the airwaves." A Family Guy logo appears on the TV set she's standing next to.
Betty White: Sure, Fox makes a fortune from advertising but it's still not enough.
Rupert Murdoch: Not nearly enough!
That particular example is more of a swipe at Family Guy than at FOX, though Matt Groening and Seth MacFarlane are on good terms in real life.
In that same episode at the end, someone calls in pledging $10,000, and Rupert Murdoch says "You've saved my network!" Bart, on the other end of the line, says "Wouldn't be the first time."
This also ended up biting them back; in the episode "Sweets and Sour Marge" they mock Butterfinger, a long-time sponsor of the show. Nestle responded by cancelling their contract. Two episodes later, "Half-Decent Proposal" aired a blackboard gag reading "I will not bite the hand that feeds me Butterfingers."
As of 2013, however, Nestle has revived the ad campaign as part of Butterfinger's "90ish" anniversary (not even Nestle knows exactly when the first bars were sold). 
In the flash-forward episode "Lisa's Wedding", set in 2010 15 years from now, we find out that all the programs on FOX have become porn. This happened so gradually that Marge hadn't noticed until that point in time.
"Marge Simpson in: Screaming Yellow Honkers": The family promotes NBC for its quality programming, ending with "How do we know if there's something good on now? Just change the channel," followed by Homer reading out a forced statement over the episode's credits that NBC sucks and FOX rules, under gunpoint, ending with him saying, "CBS: great," and being shot.
"Sunday, Cruddy Sunday" has Homer and company tangling with Rupert Murdoch, who refers to himself as "the billionaire tyrant." (Murdoch was actually playing himself.)
Another example is when a promo for Joe Millionaire goes across the top of the screen. Homer then eats part of it, but disgustedly spits out the FOX logo.
In "Marge in Chains", the Flanders' kids have been infected with the "Osaka Flu" going around town. Ned then asks himself why God has "forsaken" them only to have a flashback to the one time they watched Married... with Children (complete with sinister lightning).
Ned: Oh Maude, the network slogan was true! "Watch FOX and be damned for all eternity!"
In "Lisa vs. Malibu Stacy," the family criticizes Lisa's recent activism:
In "The Simpsons 138th Episode Spectacular" you may remember Troy McClure from such Fox network specials as Alien Nose Job and Five Fabulous Weeks of "The Chevy Chase Show".
In "Treehouse of Horror IX" Ed McMahon would like to remind you that the FOX special World's Deadliest Executions is brought to you by the producers of When Skirts Fall Off and Secrets of National Security Revealed.
In "The Simpsons Spin-Off Showcase", Troy McClure says that FOX approached the writers of The Simpsons to create "35 new shows" to fill a "few holes" in the schedule. Cue a poster of the FOX schedule: A slot each for The Simpsons, The X-Files, and Melrose Place, All other slots are question marks.note Actually, that week alone, Beverly Hills 90210, Married... with Children and King of the Hill’s ratings all outperformed The Simpsons’.
And then there's a scene in "Sideshow Bob's Last Gleaming" where Rupert Murdoch himself (speaking with a bad Australian accent) was in jail with Bob. They actually had to ask for and GOT permission from Murdoch himself for that one. His response was apparently "I would be honored to be in jail in The Simpsons."
In "’Scuse Me While I Miss the Sky," Lisa, seeking to have Springfield's lights turned off so she can see an upcoming meteor shower, complains that the only thing she can see in her telescope is the FOX satellite. The screen then cuts to a broken, falling apart satellite that's only being held up by regular party balloons.
And in "Homer Simpson, This is Your Wife," FOX is described as the home of the world's worst sitcoms, before Lisa points out that the show Mother Flippers (i.e. Trading Spouses) is a rip-off of an existing show. She is bribed with a FOX sweatshirt, but when she points out it's actually an ABC sweatshirt they throw in the American Idol holding pen.
"Behind the Laughter": We know it's off-canon, but the only reason that The Simpsons got picked up as a show was because Marge's hairdresser was also president of the FOX network.
In the couch gag for "Elementary School Musical," the 22nd-season premiere, a FOX executive appears giving the Simpsons a cupcake with a candle on it to celebrate the beginning of the season. After Maggie blows out the candle, the executive takes the cupcake and eats it himself.
Subverted in "She Used to Be My Girl"; when a media circus hits town, the Fox News van is very large and rolls into view while "We Are the Champions" plays.
The show did a parody of The Island of Doctor Moreau called "The Island of Dr. Hibbert." In it, Dr. Hibbert has been turning the people of Springfield into half-men/half-beasts. He himself comes out wearing a fox stole which resembles Mr. Burns, prompting Bart to say, "Ooooh, he got the FOX treatment."
In-universe example in "’Round Springfield". Krusty bad-mouths Percodan while being taped, then mentions "a word from our sponsor," who also happens to be ... Percodan, the very company he just criticized. Cue Oh Crap moment for him.
In "Sideshow Bob Roberts," Larry King is moderating in a mayoral debate. Before the debate, he addresses the audience.
King: I'm your moderator, Larry King. Now, a word to our audience: even though we're being broadcast on...FOX, there's no need for obnoxious hooting and hollering. *Cue obnoxious hooting and hollering*
Not to mention "The Fool Monty," for which the premise is that Fox needs to boost its ratings, and the execs decide that a new, terrifying disease would stir up the populace. However, it would be immoral to lie about the existence of such a disease. Their decision? "Let's release a real deadly disease, then blame it on something!"
And their first episode after coming back, as previously mentioned.
Revisited in the episode "Family Gay." At a horse race, the announcer rattles off the list of entrants, all of them named for recently cancelled FOX shows.
They also inverted this trope, in the episode "Death Is a Bitch." Assigned by Death with the task of killing the cast of Dawsons Creek, Peter demurs:
Peter: I'm not gonna kill those kids. If they die, I'll have nothing to watch on Wednesdays. [Glancing at the camera, and breaking out in a nervous grin] Other than the fine programs on FOX.
In "Meet the Quagmires" we get this exchange:
Molly: Hey did you guys hear on the news how President Gore hunted down and killed Osama bin Laden with his bare hands? Lois: Yeah, who woulda thought that bin Laden was hiding out in the cast of Mad TV? Quagmire: Man, the perfect hiding spot. The one place no one would look!
Also an Actor Allusion, as Alex Borstein who voices Lois had a recurring role on MadTV.
Then there was the episode where they took one potshot after another at FOX News when Lois went to work for them.
In "Something Something Something Dark Side" the opening scrawl turns into a ramble about how FOX thought so little of the Star Wars franchise that it did not bother to retain merchandising rights, handing those off to Lucas. It then goes on to point out precisely how valuable those merchandising rights turned out to be and questions the sanity of FOX stockholders for sticking with a company that makes such unbelievably stupid choices about money.
In "Big Man on Hippocampus," there was what appeared to be the trademark black with white lettering panels from [adult swim] questioning why anyone would watch the show on FOX since it is much funnier on Adult Swim.
Peter closes the episode "Three Kings" with "Now stay tuned for whatever FOX is limping to the barn with."note It was Seth MacFarlane's second TV show, American Dad!.
They've even made fun of TBS in "Hell Comes to Quahog":
From "Excellence in Broadcasting," courtesy of Family Guy wiki: After Brian said that he was moving in with Rush Limbaugh, after Stewie says "Ooh, this is a bad idea. But I guess that FOX has bad ideas, huh?", a promotional bumper for Sons of Tuscon shows up and Stewie says to the viewers pointing at the bumper, "Let's all just sit here for a moment and remember that this was a thing." note A bit of background; Sons of Tuscon was a live-action sitcom about a group of kids whose dad was in jail for insider training that hired a slacker loser to pretend to be their father. It aired during the "Animation Domination" block (despite not being animated) and often pre-empted American Dad! so it's likely this was also revenge.
In the American Dad! episode, "Less Money, Mo' Problems" the end credits has a preview for fictional show "Shoe Police" (a Call Back to a joke earlier in the episode) that is reviewed as "A new low for FOX".
The intro to the first Futurama movie is a long string of jokes where the cancellation of the show is compared to Planet Express' flight license being canceled by the "Box Network," which is in turn an unending string of attacks on Fox for canceling the show in the first place.
In the first string of lampshade jokes that opens the movie, the Professor mentions that the executives responsible for their cancellation had been fired, then beaten up, badly mauled and finally ground into a fine powder that was then packaged and sold as "'Torgo's Executive Powder,' a product with a million and one uses", which the Professor uses to powder his crotch.
Fox is repeatedly the target of jabs during the series. Such as this exchange from "When Aliens Attack":
Fry: Wow, so this is a real TV station, huh? Technician: Well, it's a Fox affiliate.
Fry then spills his drink on the control console, knocking the station off the air. The technician panics, but Fry is confident that nobody will notice.
Technician: Oh my God. You knocked FOX off the air! Fry: Pfft, like anyone on Earth cares.
The trope strikes again in the very first Comedy Central episode, which opens with a still of the Hypnotoad while a voiceover by Bender tells the viewer, on the count of three, to forget the show was ever cancelled by idiots and revived by... biggeridiots.
This is something of an inversion of this trope, for instead of mocking their old network, they mock the one they are just picked up by. They don't have a single bad word to say about [adult swim], and for good reason.
Back when they were on Fox, the crew go on a tour of Hollywood, where the tour guide says the Fox logo spotlights are used to blind pilots so that they can film the resulting plane crashes.