From what I've read, people in Northern Ireland are strongly in favour of keeping the boarder with Southern Ireland open. So that probably is going to one of the major demands of the DUP for supporting the Conservative government. 

Norway, though not part of the EU, has free trade with the EU. However they have to comply with all the rules and regulations that apply in the EU. 

So if the UK wants this too,  it sort of looks like they will be stuck with EU regulations that they won't have any voice in drawing up. 

But they may be able to control immigration, which might be what BREXIT was all about anyway.