From the information pack on Invalidity Pension
Invalidity Pension is a weekly payment to people who cannot work because of a long-term illness or disability and are covered by social insurance (PRSI).
And from the operational guidelines on Invalidity Pension on the Medical Condition:
A claimant must be regarded as permanently incapable of work, which is defined as:
Incapacity for work of such a nature that the likelihood is that the claimant will be incapable of work for life OR
an incapacity which has existed for 12 months prior to the date of claim, and where the Deciding Officer or an Appeals Officer is satisfied that the claimant is likely to be unable to work for 1 year from the date of claim.
There are no specific illnesses listed, it all depends on whether the person applying is able to work or not.