Marge makes a "Model Homer" to help remind the kids of their dad. After the model falls apart, the next shot goes outside and you see 'Santa's Little Helper' burying the Homer 'head'. They have not been given Santa's Little Helper yet.
When Marge is crying in the bedroom about Homer having to say he is unmarried, 8-year-old Lisa is clearly visible in a portrait above the bed.
When Barney is in the control of the recording studio he puts in a tape and hits the red button. His tapes begins to play, but the red button is always record.
The group sings at the local church with a sign outside displaying the group name: "The B Sharps". However at that point in the story, they hadn't come up with a group name yet. While this most likely is a simple continuity error, it could be a result of the moronic Homer telling the story and getting things wrong.
When the quartet are standing outside of Moe's discussing a name for the group, the sign on the bar says "Moe's Cavern." This is not an error; it is a deliberate reference to the Cavern Club, the club in which The Beatles got their big break (one of many Beatles homages here).
In the flashback portions, which take place in 1985, Homer still has hair. However, in the episode "Lisa's First Word" (which takes place in 1983), Homer is revealed to have gone bald by then. This could be the result of Homer's storytelling.
The flashbacks are set in 1985. Baby Lisa says that Homer, at the Grammy Awards, beat Dexys Midnight Runners. DMR's only big hit was Come On Eileen, which came out in 1983. The band had no sizable hits after that and certainly would not be a contender for any Grammy awards after 1984.
If Bart is 10 and Lisa is 8, and the flashback is 8 years prior, both children are much too old in the scene where the quartet sings at Reverend Lovejoy's church. Bart should be a toddler and Lisa should be an infant (if even born at this point). Bart appears to be about 6 or 7 while Lisa is 3 or 4.
In 1985, when Homer wins a Grammy there is a sign that reads "29th Annual Grammy awards". The 29th Annual Grammy awards were in 1987; 1985 was the 27th Annual Grammy Awards.
Even though the flashback scenes are set in 1985, when singing on Liberty Island, Homer refers to the statue as 100 years old, however The Statue of Liberty did not turn 100 until 1986.
Despit the fact that Wiggum had been thrown out of the group, he can be heard singing when they sing on the boat in New York.
Even though Santa's little Helper wasn't "adopted" until Season One, he shows up during Homer's Barbershop Quartet. This story takes place several years before he was "adopted" by the Simpsons. By the time he joins the family, Bart is 10 and Lisa is 8. But during Homer's Barbershop days, Bart is 4 and Lisa is two.
The bully who is selling fake "Be Sharps sing on rooftop!" newspapers talks with Nelson Muntz's voice, and even uses Nelson's signature "Haw Haw!" laugh. Yet his appearance is just a generic character that is never used again.