The Rolex Series event was scheduled to take place at Mount Juliet from May 28-31, with former US Open champion Graeme McDowell acting as tournament host.
"As important as the Dubai Duty Free Irish Open is to all of us, everyone's health is our only concern," McDowell said. "My thoughts are with everyone affected by the crisis and I hope everyone keeps safe and well during these difficult times."
Including majors and World Golf Championship events, a total of 12 European Tour tournaments have now been postponed or cancelled, while the Tokyo Olympics have been postponed by a year.
European Tour chief executive Keith Pelley said: "The decision to postpone the Dubai Duty Free Irish Open follows consultation with all stakeholders and was made with public health and well-being as our absolute priority.
"Our thoughts are with everyone right now and we are all united in trying to fight the spread of the pandemic.
"With this in mind, we will continue to evaluate all aspects of our 2020 European Tour schedule, and discussions on the rescheduling of postponed events will remain ongoing until we have clarity on the global situation."
The Trophee Hassan II, which is scheduled to be held in Rabat, Morocco from June 4-7, is now the next possible event on the calendar.
The US Open is due to take place at Winged Foot from June 18-21, but the course is just five miles away from the coronavirus containment zone set up in New Rochelle, New York and non-essential businesses have been closed, putting a stop to preparations.
A decision on staging the tournament is expected soon.
The R&A is currently proceeding as planned to stage The Open at Royal St George's from July 16-19 and the AIG Women's British Open at Royal Troon from August 20-23, but is undertaking a "comprehensive evaluation" of its plans and considering contingency options.
