Can I land a plane in a field in the UK and Republic of Ireland?
No, you can't shuttle back and forth over the border from one field to another between the UK and Republic of Ireland (RoI) because of regulations. The RoI requires you go through an airport of entry when entering and leaving the country, see the IAA AIP Gen 1.2-1:


  Aircraft landing in the territory of Ireland must first land at an
  International aerodrome or at a licensed aerodrome where customs and
  immigration facilities are available. Aircraft departing must finally
  depart from an International Aerodrome or from an aerodrome where
  customs and immigration facilities are available.


The UK has no requirement to take off and land from a customs airfield, you can go and land from anywhere (from a customs perspective), which is handy. You do have to notify Special Branch at least 12 hours ahead of your flight if you are going to and from the RoI, channel islands or Isle of Man and give them details on who is going and where you are going from and to. You must do this even if you plan to go to and from a customs airport on the UK side. 

In the UK you can use a field up to 28 days a year without getting planning permission from the council, provided you own the land (or have permission from the owner) and it is safe to do so. If you plan to do more than 28 days you'd need to talk to the local council and the CAA. I don't know about the RoI though.