Leaving aside insurance, it is most unlikely that UK drivers will need an International Driving Permit to drive in Ireland. Ireland does not currently require non-EU drivers to hold an IDP and that is unlikely to change when we leave.
As far as insurance goes, I believe the "Common Travel Area" rules will trump those of the EU. The CTA is a bilateral agreement between the UK, Ireland, the Channel Islands and the Isle of Man and although I have not checked I believe that UK insurance policies are recognised across the CTA without a green card.
As an aside, I believe that, despite all this hysterical hype, pragmatism will prevail and that no difficulties will be encountered post-Brexit by drivers wanting to drive in Europe.