I am assuming your accommodation is sorted out? As far as walking around the Burren is concerned then no change to that and the cliffs will still be there although the visitors centre might not be open. Ireland generally closes down from the 24th - 27th. Christmas day is definitely a closed day and St Stephens (26th) is a main holiday. Trips to the Arans' are already off until April but with a car you will be ok to trip about, at least the roads should be quiet although I am hoping the weather will be better than last year (white Christmases are pretty and ok for the kids but a pain out on the mountain).
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Two rules for Ireland, Take your time & bring a sense of humour
As Tony pointed out there will be several days when things are closed down in Ireland for the holidays. However, secenery is always open. From Doolin some outdoor activities and/or driving day trips:
Cliffs of Moher Bridges of Ross Loop Head Burren Connemara Lough Derg
It can be very windy, cold and rainy at that time of year so take appropriate clothes. Also, very short days. Darkness can arrive by 4 or 4:30 in the afternoon. Plan accordingly.