Visible crew/equipment: When Jack opens the medicine cabinet to get the shaving cream, in the mirror's reflection you can see the camera as well as where the set ends.
Good Old Reliable Janet - S3-E2
Continuity mistake: When Janet and Mrs. Roper are found nude behind the bushes, Larry leaves to get towels. However, the yellow towel suddenly appears briefly on screen several seconds before Larry returns.
Double Date - S3-E1
Visible crew/equipment: At the start, while Mr. Roper is in the kitchen the faucet sprays water and the doorbell rings. Note the cabinets and wall behind him. When Roper walks out of the kitchen, as the swing door is open we can see the kitchen, and the upper cabinets are split apart with the wall missing, so the area behind the kitchen set is visible. (00:00:50)
Visible crew/equipment: When Mr. Roper gets the idea that it's Jack's diary, Jack tells him not to worry and it'll be their own little secret, then in the next wide shot we can see the bank of set lighting at the bottom of the screen. (00:21:15)
An Anniversary Surprise - S3-E20
Visible crew/equipment: At the end of the episode when Jack falls back off the chair, a boom mic is very visible above him. (00:25:15)
Plot hole: Janet and Chrissy accuse Jack of making the fake BBQ telegram. However, Jack was at The Apartment the whole time, never left once until after he found out Lori was the one who made the telegram. If it was Jack who made the fake telegram, he would have had to leave The Apartment to do that.
Suggested correction: True, there's no logical way that Jack could have sent the telegram. But Janet and Chrissy were upset about it - when someone's upset, they don't always think in a logical way.
Or he could've called Larry and had him set it up but I don't think he ever used the phone to call anyone.
Good Old Reliable Janet - S3-E2
Character mistake: At one point in this episode, Jack is chatting with Larry about the nude beach and in frustration says "all the best looking girls were wearing moth-eaten blankets" right in front of Mr. Roper as well as showing interest in seeing Janet without her clothes also in front of Mr. Roper. Jack is supposed to pretend he's gay when Mr. Roper's around him. Roper luckily didn't pick up on it this one episode either time (rather showing concern for his wife).
Plot hole: At the party, when Jack tries to suddenly stop many guests from eating his tainted cake, he knocks away the cake from many people to prevent them from eating it - but at least two minutes or so have gone by since the guests were given the cake slices so most likely, many (if not all) the guests would have tried the cake within those two minutes. And yet somehow Mr. Roper was the only one who is revealed to have actually tried the cake in all that time.