The lack of a contract isn't really a problem, your tenants would be considered to have a Part 4 tenancy
http://www.citizensinformation.ie/en..._to_leave.html
To evict them on the grounds of breach of obligations (to pay rent), you first need to send a 14 day notice requesting the sum due.
Should they not pay, you may then issue a 28 day notice of eviction.
Actually, in this case the lack of a fixed term contract does make a problem.
If there is no fixed term tenancy, then there is a Part 4 tenancy. With a Part 4 tenancy, TWO notices are required before a Notice of Termination for rent arrears can be valid.
A first Notice pointing out that the rent is in arrears and should be paid immediately. There is no template for this notice nor is there a time frame. Therefore, a day or two later, a 14 day notice may be served on the tenant. Only after a tenant fails to pay all outstanding rent may a valid Notice of Termination be served.
However, if there was a verbal agreement that the tenants would stay for a fixed period of time, then a verbal fixed term lease was entered into. However, if a case was brought to the PRTB, it would be a matter of one party's word against the other's.
If the landlord can produce a copy of the agreement that was to be signed was a fixed term contract, this would help the landlord but would not by itself be conclusive evidence.
If a fixed term agreement (verbal or written) was in existence, then only a 14 day notice of arrears of rent needs to be served before a Notice of Termination.