Not sure what you mean by this.
In most respects Britain outside of the EU would interact with Ireland in much the same way. Certainly in respect to the movement of people, as has been pointed out in this thread already. It’s also worth remembering that there are around 300,000 people from the UK who live and work in Ireland as well, so any attempts from the UK to change things around would effect those people as well, and that would not be a good idea politically - to put it mildly. In terms of other things like social welfare, voting in elections, things would also remain the same as they are now because these agreements are not based on the EU.
The only big change would be trade, but given that both governments on either side of the Irish Sea are economically liberal (and will likely remain so after the upcoming elections), I can’t foresee much problems in quickly hammering out a free trade agreement. The UK has had a problem with our low rate of corporation tax in the past, so that might be the only sticking point.
I think Ireland mostly wants the UK to remain in Europe because we have similar views on how Europe should be run. Ireland does not want to see a european super state any more than the UK does, so it’s important to the small like minded nations (the Dutch, Estonians, etc.) to have someone like Britain around to keep a check on the French and the Germans.
Ireland could actually see some benefits from the UK leaving though, it would make a very good entry point for the UK market for european companies for example - much like US companies use us for in terms of europe generally (and even more so for Scotland in the case of independance there).
