Liability to UK tax will depend on 2 things - first, his residence for tax purposes. If your dad left the UK intending to move to Ireland on a permanent basis, and his return visits have not exceeded 90 days a year on average, then he will probably be deemed non-resident for tax purposes. Secondly, if he were indeed non-resident for tax purposes, he would only have a liability to UK tax on income arising here (such as a pension, or investment income).
I do not know the Eire tax laws, but would assume that once he arrived there to live, he would become liable to Irish tax on his worldwide income. There is a double taxation agreement between the 2 countries, which would ensure his UK income wasn't taxed twice.
So the issue is what sort of income we are talking about, is it income that has been taxed at source?
The UK tax authorities can routinely ask for tax for the previous 6 tax years, and can go back further if they suspect deliberate non-declaration. I don't know the position in Eire, but again, would expect something similar.
I would suggest he tries to sort this out now, otherwise the mess will get worse, and it may be, if the amounts are small, that it isn't as bad as he thinks.