It is incorrect to say that Tut was Irish. It is more correct to say he has a genetic relationship with the people who eventually settled in Ireland.
Agriculture apparently originated somewhere in the Levant. It would not be surprising if proto-European agriculturalists came down the eastern Mediterranean coast, found the rich soil of the Nile valley, and settled there, displacing any original inhabitants.
When PBS was doing a program they scanned the camera over Tuts DNA. Someone recognized it as Irish. More research revealed that Tut was related to half of present day European men. Another article said that Tut’s parents were his brother and sister. Lots of strange things.