If your estate consists of Shares or bank acccounts etc. Probate will have to be taken out in the country where these assets exist. Therefore the Executor of your estate would most likely have to take out probate in the UK and Ireland.
I know someone who inherited shares in the US and it was actually going to cost more to take out Probate there than the shares were worth, so they never did anything with them.
However, whoever inherits it will only have to pay inheritance tax to one government as at this current moment in time there is an agreement on that.
However, with any luck it's a long ways off and who knows what way thing will change. Obviously the people named in your Will would have to look at the Inheritance Tax laws of both countries when the time comes and opt for whichever would be most beneficial to them.