I suspect you think he isn't liable to tax in Ireland as he is employed by a UK company and is paid into a UK bank account?
As Irish resident he is taxable on foreign employment to the extent that his duties are performed in Ireland. As he works from home, in Ireland, it would appear he will have an Irish tax liability.
It is the responsibility of his employer to operate Irish PAYE on any payments to him. If he has paid UK tax he may be able to get credit for this.
Having said all that, TR1 is for registering for self-assessment (mainly self-employed). I can't see why he would need to complete this.
You should really seek professional advice on this.