Taoiseach Leo Varadkar said he and the outgoing Prime Minister had got to know each other very well in the last two years.
After Mrs May announced she will step down as PM and Tory leader on June 7, Mr Varadkar said: “She is principled, honourable and deeply passionate about doing her best for her country, and her party.
“Politicians throughout the EU have admired her tenacity, her courage and her determination during what has been a difficult and challenging time.
“Theresa May strove to chart a new future for the United Kingdom. I want to wish her the very best for the future.
“I look forward to working closely with her successor.”
Micheal Martin, the leader of Ireland’s main opposition party Fianna Fail, said it is a “difficult day” for Mrs May.
“She has committed her life to public service and has had responsibility for leading the very highest offices in British life,” he said.
“I wish her well for the future.
“Her fate is a reflection of the emerging and ongoing crisis in British politics as a result of Brexit and is a reminder of how unstable and potentially damaging this process remains.
“The coming leadership election within the Conservative Party has the potential to further destabilise the Brexit process.
“In Ireland, those of us entrusted with positions of leadership must remain vigilant and stay alert to the threat of a no-deal Brexit.
“Our hope will be that her replacement is someone with the skills and determination to achieve the compromise needed to allow the UK and the EU to move on.
“We must also ensure that this development is not used to further delay the restoration of the Executive and Assembly in Northern Ireland.”