Christmas is thoroughly celebrated throughout the island and while many pubs, restaurants and hotels will indeed be closed on the 25th, there will always be some that remain open for this special day. I'd advise you to book your accommodations
well in advance, as these days tend to sell out quickly. Some hotels and B&B's offer Christmas dinners for their guests or can recommend a nearby restaurant that will be open.
You'll find there will still be plenty to do around this time of year. Places in Dublin and key attractions remain open to visitors, while most major cities and bigger towns will have their own Christmas markets and festively lit-up streets. Our hub page on Christmas in Ireland
is a great place for some information and inspiration on what to do. Amy and Rena came to us with similar questions
some time ago on this Community. There are lots of useful tips to find there as well!
As for the weather, it will probably be cold, but rarely below freezing. Bring something against possible rain showers and you both should be more than fine. :)
Have you a place in mind that you'd like to visit or are you thinking about travelling around?