What does "You are not Irish" mean?
Jack's mother will be so relieved that none of the other stories about Anna are true that she will be less upset when she finds out Anna is not Irish.
The context allows one to assume that Jack's mother has her heart set upon any woman Jack ends up with being Irish. Since Anna presumably isn't, Jack has told her all sorts of horrible untruths about Anna, painting her as the worst sort of person imaginable. The supposed result of her finding out that Anna is nowhere near as bad as she had been described is that she will be so relieved, she will overlook her non-Irishness. The joke could be told of any nationality or religion; the Irishness is incidental
One thing a non-native speaker such as yourself might not notice is the names.
Delaney is an Irish name. Petrescu sounds Eastern European to me. I'd guess Polish, if forced to guess.
So this is a bit of a joke, in that he's telling her that he softened his mom up for the big blow of his girlfriend not being of the same ethnic background as his family, by telling a bunch of lies to make her sound like a moral reprobate. The idea being that when she finds out those things aren't true, the not being Irish part will seem like small potatoes.
The list of lies were also a bit tailored to an Irish (or at least Catholic) mother's list of bad qualities in a person. A person who has been married more than twice is almost certainly divorced at least once, and devout Catholics don't believe in divorce. Also, there's an implication there of out-of-wedlock sex ("not necessarily by three husbands"), which devout Catholics also frown heavily on. Such a woman would not be allowed to take Communion in many Catholic churches. If he married her, there's a good chance he and their kids would not be welcome either, because the Church would not consider their marriage valid. It would be like her son was signing up for excommunication.
Yes, it makes perfect sense if you replace Irish with pretty much any other USA immigrant Catholic community (Italian, Polish if you change the girlfriend's name too, etc.). Japanese would probably work too, but you'd have to change the list of lies to something a stereotypical Japanese mother would find repellent in a person. I doubt English or Scottish would work very well, as those are the two oldest (one might say founding) immigrant communities in the USA, so they tend to assimilate with few issues.
Delaney has told his mother a large number of highly negative untruths about his girlfriend ("you work in a far older profession than art consultancy," refers to prostitution, the world's oldest profession, as do all the other lies, about multiple husbands and children). He has also told the mother (Delaney, an Irish name),* one "positive" lie, that she's Irish when she really isn't.
After the mother gets the feeling of relief by the "fact" that the girl (Romanian, by nationality) is none of the bad things she's been told, the man will then confess that the part about her being Irish is also a lie, to soften the blow.