Plot hole: They're in the Clam complaining that they can't drink when they make the revelation that Brian can because he's technically 56. But Quagmire should be able to drink, as in a much earlier episode, he's revealed to be over 60.
Suggested correction: Quagmire considers the rest of them to be his best friends. He isn't drinking because they can't drink. He is complaining because since his friends can't drink - he won't drink, so he is still pissed.
It's still a plot hole considering Quagmire's age. I think the writers messed up big time.

A Very Special Family Guy Freakin' Christmas - S3-E16
Other mistake: When Quagmire is sticking his body out of the car door and yelling "Whooooooa! Whoooooooa!", look at the person in the back seat. It certainly isn't Joe, who was there in the previous shot, in fact it looks like another Quagmire. (00:05:30)
Suggested correction: That most certainly is Joe. It does not look like Quagmire at all.
While it's hard to call the face another Quagmire, the person in the back is wearing a Santa hat and same color/style coat as Quagmire. Joe was wearing a grey knit cap and different color/style coat. The person in the back also has black hair like Quagmire and not the brown hair Joe has.
Continuity mistake: In the first shot of the flashback showing the family watching "Philadelphia," Peter's eyes are about twice as big as they normally are. They're back to normal in the next shot.
Suggested correction: In this scene Peter's eyes are supposed to be larger then normal. The reason for this is he is explaining what happened last time he drank Irish Coffee. The large eyes are used to show that he is "buzzing" from the drink, as you will also notice that he comments that Tom Hanks is a funny guy and when the character says "I have AIDS" Peter bursts out laughing.
The mistake is a continuity error regarding Peter's eyes going from big (as a result of the Irish coffee) to as they are usually in the next shot. The mistake still stands.
Mr. Griffin Goes to Washington - S3-E3
Continuity mistake: When Mr. Weed is crying about losing his job, we see a view of a long line of trees in the window behind him. The camera switches to a close-up of Peter. When it zooms out again, the trees on the right of Mr. Weed are missing.
Suggested correction: I watched this part of the episode twice and no trees are missing in the window behind Mr. Weed after the close-up on Peter. Actually, Mr. Weed's head is covering part of the window where you can't see the trees on the right when he's lying on the table, but when he gets up you can see all the trees.
Continuity mistake: During the scene where Chris walks into a hardware store, the antennae on the control on the back of his head change from straight to curled to straight again.
Suggested correction: I watched the hardware store scene carefully but I didn't see any difference on the format of the antenna on Chris' head. It's always stay curled in the whole scene.

I Am Peter, Hear Me Roar - S2-E8
Revealing mistake: When Brian says "Well face it Peter, your attitude towards women isn't exactly enlightened", Brian's dog tag is the same color as his fur. It also stays still, even while Brian is moving his head. (00:11:10)
Herpe, the Love Sore - S12-E16
Factual error: Stewie could not have gotten herpes by blood - it's only transmitted by infected saliva or skin.
Suggested correction: Please tell me that you are not a doctor! The two types of herpes viruses, HSV1 and HSV2 can be transmitted in blood as well as saliva. Such transmission is very rare as the virus deactivates rapidly in stored blood, but it is incorrect to say that it cannot happen.
The Cleveland-Loretta Quagmire - S4-E5
Plot hole: Brian and Peter see Loretta and Quagmire having sex yet, they aren't sure who she's having an affair with. There's no way they couldn't have noticed it was Quagmire - the couch was facing the front door! And obviously Quagmire was on top of Loretta considering the fact that he leans up, says something, then she tells him to get back down here and finish.
Other mistake: When the family is driving to New York, another red car passes in front of them from the right where there is no road. (00:09:35)
Suggested correction: Usually this would be right, but it's evident by their positioning that Peter is actually driving his car right in the middle of the road on the dotted line and not staying in one lane. This is why the truck passes on the left to get in front and then the car on the right, both also getting centered on the dotted line.
Factual error: Joe's plan to trick OJ into confessing to the 1994 murders is moot since, having been acquitted, the double jeopardy provision would prevent him from being tried for the same crime twice, and as a cop, Joe would know that.
Suggested correction: The plan to get OJ to confess had nothing to do with getting the confession, so he could be arrested or tried again. And at this point, the civil trial already found OJ "responsible" for the murders. Since OJ has always maintained his innocence, getting a confession on tape would just be big news.
Blue Harvest - S6-E1
Trivia: When an explosion on the Death Star sends an Imperial stormtrooper flying, the stormtrooper lets out a Wilhelm scream.
Suggested correction: I'd say given that all Star Wars movies use the scream, it would be unusual for it not to appear.