DIRT stands for "Deposit Interest Retention Tax", so called because it's deducted by the deposit-taker and retained by it, so you never see it. All you get credited to your account is the amount of interest that remains after DIRT has been deducted.
You are also liable to income tax on the (gross) interest paid but (a) you get a credit for the amount of DIRT the deposit-taker has retained, plus (b) a generous Oireachtas has decreed that, if your interest has been subject to DIRT, your liability to income tax is capped at the amount of DIRT deducted, so you never have to pay any more than has already been deducted.
But deposit-takers outside Ireland are not subject to Irish tax law. They don't deduct DIRT from the interest they credit to your account. DIRT doesn't apply to interest earned on deposits in other countries.
Income tax does, of course. So if you're an Irish resident with money deposited abroad, the interest you earn is not subject to DIRT, but it is subject to income tax.
To avoid any suggestion that deposit-takers in other member states are being discriminated against, if your deposit is in another EU member state the amount of income tax is set at the amount of DIRT that would have been levied, if the deposit had been in Ireland.
But concerns about discrimination don't apply to non-EU deposit-takers. So if your deposit is with, say, a US institution, the amount of income tax levied on your interest is (a) income tax at your marginal rate or, if higher, (b) income tax at the rate at which DIRT would be charged, if the deposit were in Ireland.
The key message here is that, though tax on interest earned abroad may be charged at the same rate as DIRT, it is not actually DIRT; it is income tax.
And, once you understand that, it can be seen that, yes, if you are non-domiciled the remittance basis does apply. Interest earned that has not been subject to deduction of Irish tax at source is taxed under Schedule D Case III. Foreign interest is not subject to the deduction of Irish tax at source so it's taxed under Schedule D Case III. The remittance basis applies to income that it taxable under Schedule D Case III. Hence, if you are non-domiciled, earn interest abroad, and do not remit that interest to Ireland, the remittance basis should be available to you and the interest should not be subject to Irish income tax. (It may of course be subject to tax in the country where it is earned.)