The most prominent likely Irish character is Seamus Finnegan.
It's never outright stated that Seamus Finnigan is Irish, but it's heavily implied.
Both "Seamus" and "Finnigan" are Irish names. Having an Irish first name by itself could simply mean his parents liked the name, and having an Irish last name by itself could just mean that someone in his ancestry was Irish. Having both an Irish first name and last name, though, implies that he has Irish ancestry, and that his parents feel connected enough with that to give him an Irish name as well.
In addition, when Professor Binns forgets his name, he calls Seamus "O'Flaherty", which is also an Irish name. This somewhat implies that Seamus is memorably Irish.
Seamus also speaks in a way that could certainly be Irish. He refers to his mother as his "mam". Some of his speech patterns could also be informal British, but it does lend support to his possibly being Irish.
“I’m half and half,’ said Seamus. ‘Me dad’s a Muggle. Mam didn’t tell him she was a witch ’til after they were married. Bit of a nasty shock for him.” - Harry Potter and the Philosopher's Stone, Chapter 7 (The Sorting Hat)
His Quidditch team support.
His favorite Quidditch team is the Kenmare Kestrels, a team from Ireland. He hangs a poster of them above his bed. In addition, while all the British wizards we see at the Quidditch World Cup are supporting Ireland, Seamus keeps his shamrock and pins it to his headboard at his dormitory.
Helga Hufflepuff may be Welsh, but there's less support for this.
In the Sorting Hat's song, he says that Helga Hufflepuff is "from valley broad". There are many valleys in Wales, so this is commonly thought to imply that Helga Hufflepuff is from Wales.
Pottermore information may imply Remus Lupin is possibly at least part Welsh.
Remus Lupin's father met his mother on a visit to a Welsh forest. She was working in an insurance office in Cardiff at the time.
On an investigative trip into a dense Welsh forest in which a particularly vicious Boggart was supposed to be lurking, Lyall ran across his future wife. Hope Howell, a beautiful Muggle girl who worked in an insurance office in Cardiff, had taken an ill-advised walk through what she believed to be innocent woodland.
This could imply his mother was Welsh, although it's not clear whether she actually was from Wales or if she was just working in Wales. The same thing applies even more to his father - he was going to the Welsh forest (and is later mentioned as having knowledge of Welsh boggarts) but he doesn't have to actually be from there.