Lufthansa
Lufthansa will allow you to take a dog from Dublin to Frankfurt in the cabin (€60 charge) but they will not allow you take the dog back FRA to DUB because (I'm serious) Lufthansa considers Ireland to be part of the UK!
The department of Agriculture, Food and the Marine here in ROI confirm there are no restrictions bringing your pet dog or cat into Ireland from another EU country (providing it has been microchipped, rabies vaccinated, tapeworm free and has an EU pet passport).
How do we give Lufthansa a polite lesson in geography (and possibly history)?