To evict a tenant for non payment of rent, you must go through the correct process as laid down in the Residential Tenancies Act 2004.
If brief, you must first issue a 14 day rent arrears notice in the prescribed format (sample with notes on the PRTB website) to give the tenant the chance to pay the rent. If the arrears are not paid then you can issue a notice of Termination to evict the tenant. Again, this notice must be in the prescribed format or it will be invalid. If the tenant has a Part 4 tenancy, two notices of rent arrears must be issued, though the first does not have to give the tenant 14 days to pay.
As regards recovering any rent arrears when the tenant has vacated, you will have to have the tenants new address or the legal documents cannot be served on the tenant. The PRTB are not very helpful in this matter. The time period may be anything from 6 months (if you're very lucky) to a year or two.
In the case where the tenant overholds, the PRTB say that they will give priority to such cases - but I wouldn't hold my breath for an adjudication.
If you do not proceed via the correct route with valid notices (even if you believe the notices to be valid - and I have seen instances of solicitors issuing invalid notices) - then it could cost you very heavily in fines of possibly up to 10,000 euros.
I hate to say it, but not receiving rent is one of the main reasons why so many landlords will not accept tenants on SW - which is a pity for the good SW tenants. Furthermore, the tenants/social welfare office seem to be able to hide behind the data protection act and not help landlords in recovering rent arrears.