Claiming noble title of Earl in Ireland that seems to have died out?

Short version: Almost certainly not.

Long version:

To find out the rules of inheritance for a Irish peerage*, you would need to check the letters patent. These will specify how the inheritance takes place (the limitation on the remainder).

For the most part, the inheritance is only through legitimate male heirs (heirs male of the body). This means descendants of the original holder only - so it is possible for it to pass back to a brother, uncle, cousin, etc. if someone dies without legitimate sons, as long as they are descendants of the man the title was originally created for.

In some cases a peerage is created with a special remainder, e.g. in the case of Earl Roberts, his sons had already predeceased him, therefore a special remainder allowed his daughters and their male heirs to inherit (there were no male heirs, and the title is now extinct). In another special case, that of the Duke of Marlborough, the rules of inheritance were actually changed by Act of Parliament to stop the title becoming extinct.

In a very few cases instead of "heirs male" the inheritance is through "heirs general" - i.e. women can inherit. The only known extant Irish peerage that descends through heirs general is Viscount Massereene (and the associated title Baron Loughneugh).

In principle, there might be a legitimate heir to a peerage that was thought to be extinct. This happened in 1831 where the title of the Earl of Devon was successfully claimed (it had been dormant since 1556) by a distant cousin. In practice, I don't see it happening. It would not be enough to show you were descended - even if you descended through the male line - you would also need to show there were no other legitimate claimants. Just like Prince Harry being in line for the throne, but not at the head of the queue ;).

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*the status of the "Irish Peerage" is a little complicated these days - the Republic of Ireland does not confer titles or permit citizens to accept them without permission. In practice this means that these titles are under the control of the UK crown, and in the Republic of Ireland they are considered 'courtesy titles' only.