You can't participate
directly[i in the Irish lottery. It's illegal to sell tickets for it in the UK.
However bookies can take bets on anything they like so, unsurprisingly, they take bets on the outcome of the Irish lottery, both in their shops and online. (The site in AC's link is that of a bookmaking service, not that of the Irish lottery itself).
From the [i]official] website:
"Do I have to be a Republic of Ireland resident to open an account?
Yes, in order to register and create an account, you must be a Republic of Ireland resident"
https://www.lottery.ie/en/Play-Online/Play-Online-FAQ
Most major UK bookmakers allow you to place bets on the result of the Irish lottery, e.g:
http://www.ladbrokes.com/ppc/sports/en/irish-lottery/
http://static.williamhill.com/sport/irish-lottery/