Stede Bonnet would've tried asking the Spanish for his sugar back, but his Spanish is - as he puts it - "very evil."
While disguised as Duncan Walpole, a bonus objective has Edward taking Refuge in Audacity by pickpocketing the three Templars huddled around a table discussing their plans, for no apparent reason. Given that he's already been winging it up to this pointnote with generally noncommital answers or silence in conversations, and improvising his Hidden Blade killing moves, that is hilariously pushing it. Moreover, due to the history of the "full synchronization" bonus objectives, this means that Edward Kenway actually did this.
You could also look at the fact that Edward Kenway was inducted into the Templar Order firstnote even if it was under the guise of the actual defector, Duncan Walpole, Edward isn't a former Assassin so for much of the beginning of the game you're once again playing a Templar surnamed Kenway... or the fact that the Templars are just as prone to self-deception-by-assumption as the player may have been to assume them to be Assassins!
Other than the attempted Assassin attackers in Havana, Edward's first subsequent encounter with the Assassins has him evading Assassin guards on his way to a temple to find James Kidd... whereupon he promptly mistakes the Assassin Order for "Monks" of a "daffy Religion".
For some added laughs, turn on the subtitles during this chapter. It turns out the Assassins knew Edward was there the whole time and are just humoring his attempts at stealth.
When a certain character reveals their true identity, she (James Kidd/Mary Read), threatens to castrate Kenway if he tells anyone. She has a very peculiar smile on her face as she moves out of the frame.
Upton: The church! In the name of the father, the son, and the holy- (beat) Mate, the father the son, and the holy- what's the third one again? Edward: Let's just keep moving, yeah? Upton: The father, the son, and the let's keep it move- no, that's not it. Edward: What does it matter?
When Edward finds Rhona Dinsmore, who's hiding under fire by hostile forces:
Poor Stede Bonnet gets the full Blackbeard treatment in one part of the game, to the point that the entire screen actually gets darker as Thatch snarls in his face with those burning wicks coming out from under his hat. Then:
Thatch: Give your Quarry something to fear, some hellish Thing from a fever'd Dream, and Men will drop to their Knees, pleading for their Lord before aught else!AAARRRRRGGH!
Bonnet: [terrified as Thatch looms over him] Oh... o-oh God...
Thatch: [nonchalantly, walking away as the screen returns to normal] 'Course it does.
There is an achievement for getting drunk off your ass and waking up in a haystack.
The in-game Abstergo emails and promos and information on their products are hilarious for being constantly Entertainingly Wrong. There's also the fact that they reject Altair, Ezio and Connor's stories as being not propaganda friendly. One can imagine that in some Assassin afterlife, the three of them are all smug about being too badass for the Templars while Aveline would be pissed off at being made a Token Enemy Minority for the public.
What sells it is the really bad American pronunciation of the names by the Abstergo analyst. This leads to gems like "Alltare Iben Lahad", "Enzio Owdeetory", and "Ratonagayden/Ratonatankon".
Fun fact: The Abstergo analyst is voiced by none other than Roger Craig Smith, Ezio's voice actor.
Thatch's excited and obviously mood-lightening announcement about his "retirement party" is oddly adorable for the guy who made it his business to be the scariest guy on sea. He unfortunately dies a couple scenes later.
Vane's robbing of Edward's oysters when they're stranded while waggling a stick around. Made even better by Edward's annoyed reply that there is plenty of food if he just tried looking. The entire mission is the first time the franchise dropped its historical dimension and devolved into pure slapstick comedy, with Vane and Edward essentially acting like a pair of toddlers fighting for room in a sandbox.
Another, in which the Jackdaw turns into the Titanic.
There is a glitch that will occasionally manifest itself where Edward will get his teeth stuck in a Slasher Smile-like grin. You can see a version of it here.
When Rackham betrays Vane and Edward, he maroons them on a damaged ship. The conversation between the two is bound to get a few giggles.
Vane: I'll gut you, Jack Rackham! I'll open you up, I'll tear out your organs, and string a bloody lute with 'em! Edward: Stop your god damned howling, Vane! It's no bloody use! Vane: Well, well, the fearsome Edward Kenway speaks! Pray tell us, captain, how to quit this predicament! And tell us what genius you have for sailing a boat with no sails and no rudder! Edward:SHUT! YOUR! GOB!
Somehow, this troper finds the "Press any button to rattle the cage." funny. Even the UI lampshades this, replacing all your action commands with "Rattle".
Abstergo's final trailer is already pretty funny on account of being Entertainingly Wrong and Stylistic Suck... but one phrase in particular stands out: "Nothing is sacred, and everyone is committed." It appears the Templars have been secretly jealous this entire time.
Early in the game, Edward runs into an Englishman at a tavern, who he immediately proceeds to insult:
Drunken Sailor: Fancy meeting a Welshman deep in Dago country. I'm English meself. Biding my time 'til the next war calls me to service.
Edward: Lucky King George, having a piss-pot like you flying his flag!
While Edward and Charles Vane are discussing tactics in "The Fireship", Rackham is sitting atop several barrels of gunpowder...smoking a pipe.
Vane: (walks over and smacks Rackham around) Bloody hell. You'd raise a cloud here, ya lout? You're sitting on enough gunpowder to blow New Providence off its rocks!
Rackham's reaction doesn't hurt:
"Rackham: Lay off me, mate! I'm well chafed!"
What sells it is Edward visibly rolling his eyes, wondering what did he do to hang out with an Ax-Crazy psycho and a drunk incompetent.
During the mission "Diving For Medicines":
Hornigold: Remember, Thatch, keep your cannons corked!
Blackbeard: Piss off, Ben!
It can be quite amusing when a taken-over Fort takes down enemy ships, especially pirate hunters.
Amusingly, by the end of the game, Edward has become an English nobleman.
In Freedom Cry, Adewale, of all people, has one of these moments when he starts gleefully gushing about the scientific benefits of Godin's expedition. His enthusiasm makes him downright Adorkable. It's a pity that he gets brought down only a second later by a What the Hell, Hero? moment.
The sound effect that plays when you rescue a shipwrecked sailor is the same sound effect you get for getting money or loot.