There is NO check at all....ever......(except when there was the foot and mouth outbreak...but even then they were totally unconcerned with nationality).
Sorry but that is false information as being tied up to some tourist travel boards will show.
Irish Immigration know this is a method being used to subvent immigration controls and do check trains and buses occasionally and have been known to stop and check cars. Majority of time you are ok BUT doesn't help if you are the exception.
Taking a copy of the passport certified as being a true copy and talk to the Irish Embassy / Consulate explaining the position, yes it may mean that you get stopped at Dublin Airport BUT if you have correspondence that shows the position from the consulate then at least they can refer to that.
I don't believe there is that much of an issue with Illegal US immigrants going into Ireland.