When Anna and Jeremy are at the restaurant, Jeremy says they "might have to go in through the spine." No surgery of any kind ever goes "through the spine," particularly any cardiac surgery. In human anatomy, the common term "spine" actually is the vertebral column. The statement made by Jeremy literally would mean you would split the patient's vertebrae to get to the heart.
When they are sitting at the dinner table with the two couples after the husband pours everyone a drink the bottle is sitting on the table with the rear label facing the camera, the shot cuts away and when it returns you can see the name of the whiskey they are drinking because the front label is now facing the camera
When Declan throws his pillow onto the bed at the Bed and Breakfast, the trim of the pillow case is at the edge of the bed, but when the camera angle changes, the trim of the pillow case is at the center of the bed.
When Anna and Declan are in the lobby, Anna's suitcase and purse are next to them (it can also be seen next to Anna when the camera pans to the man taking video on his phone). When Jeremy and Anna leave the lobby, neither of them have the case or purse, meaning they must have been left behind. Considering how attached she was to it, it's doubtful, no matter how flustered she was, that she'd leave it unattended to go up to the room.
While crossing the Celtic Sea, the captain of the boat says that because of the storm they would have to put in at Dingle instead of Cork. They would actually have to have passed Cork and sailed for several more hours to reach Dingle, which is on the West coast.
Anna calls Jeremy (in Dublin) from the B&B at approximately 8:30am on a Sunday morning. He tells her that he just got off the phone after learning that the apartment was theirs. With a 5 hour time difference it is highly unlikely that the deal was closed in Boston at 3:00 am - on a Sunday, no less.
When flying from Boston to Dublin, Anna's flight is diverted to Wales. Unless she had a layover in London, the more likely diversion would be to Shannon Airport in the West of Ireland. She then takes a boat from Wales to Dingle. Again, unlikely since Dingle is on the west coast of Ireland.
While in Dingle, it is shown to be a tiny village, which it is, but not as small as depicted. Also, several shots are shown on the Cliffs of Moher, north of Shannon, whereas Dingle is south of Shannon.
The whole premise of the film is that on Leap Day (i.e. February 29th) a woman can ask a man to marry her in Ireland. Yet when they finally arrive in Dublin there are leaves on all the trees in the park and people are going around without any heavy coats. Either it is a freakishly warm winter or the filmmakers shot the scenes in Dublin during the summertime.
The carrots that Anna picked from the garden were not the same carrots that she was chopping in the kitchen in the next scene. The garden carrots were rather small and thin. The carrots in the kitchen were much larger and thicker.