Hi, we travel to Eire a couple of times a year and have done so for the past 25-ish now.
This is the official line:
"The requirement is that all pets travelling from the Republic of Ireland to the UK should be microchipped, vaccinated against rabies and accompanied by a pet passport. However, as both countries have had no indigenous rabies for many decades compliance checks on pets travelling between them will not be applied." BUT we have been told by the Ferry companies that DEFRA were threatening spot checks and while lots of people do still travel without the passport, for me, it just wasn't worth the risk, so now both my dogs have their passports sorted.
Not, all they need is the rabies vaccination, my vet double checked just in case and tapeworm treatment isn't required between the UK and Eire as they have the same status.
Here's the quote and link for that:
"Tapeworm rules apply to pet dogs only. Not less than 24 hours and not more than 120 hours (1 to 5 days) before its scheduled arrival time in the UK under the Pet Travel Scheme, your dog must be treated against tapeworm and the treatment recorded in the EU pet passport or the third country official veterinary certificate.
No treatment is required for dogs entering the UK directly from Finland, Ireland, Malta or Norway."
Defra Site Link and Gov.uk Site
The list of dogs covered by Eire's DDA covers more breeds than the UK act and applies to:
American Pit Bull Terrier,
Bulldog,
Bull Mastiff,
Dobermann Pinscher,
English Bull Terrier,
German Shepherd (Alsatian)
Japanese Akita,
Japanese Tosa,
Rhodesian Ridgeback,
Rottweiler and
Staffordshire Bull Terrier and
... every dog of the type commonly known as a Ban Dog (or Bandog) and to every other strain or cross of every breed or type of dog described.
Completely ridiculous. 
Basically they have to be kept on a lead of no longer than 1m, muzzled and not held by a person of younger than 16 years old when in a public place.
More info can be found here.
Having said that, I travel regularly and have seen most, not all, of those breeds running free on the beach quite happily, so it's not strictly enforced, at least rurally. Not sure if that would be different were you to try and walk one of them around a town centre and it's worth noting that, at least in the area my family are from, people tend to be wary of large and well-muscled breeds of dogs in general.