You can rent a car or van and drive it with your things and dog to Ireland and hand it in at a location of the rental company, if you have rented it from a main company that has locations with vans in both Italy and Ireland. That is if you can find a company renting out vans with locations in both countries. And when you do you will have to pay a huge drop off fee as they will have to have a professional driver drive it back, as it has the steering on the wrong side for rentals in Ireland.
It may well be cheaper to rent the van a little longer, drive it back yourself and hand it in where you rented it or at least at an other location in Italy and fly to Ireland.
Likely as cheap, easier and more convenient to buy a car or van in Italy while you still live there and drive it to Ireland. It will have the wrong license plates to sell (well) in Ireland but depending on the kind of vehicle and its value, you may sell it for what you paid for it minus what a rental van would have cost you.
And if you are really lucky you might even be able to sell it for more than what you paid, if you happen to find a van that is more popular in Ireland than in Italy.
Traveling with your goods and dogs by public transport is likely a no go. You would likely have too much to transport to take the train.
Dogs are not always allowed on trains and most important, dogs are not accepted on most (or even all) ferries if you are a foot passenger. Even if it is possible, it will not be an easy travel and likely not cheap either.