An Irish tan occurs when a very pale-skinned Irish person gets too much sun, causing their skin to become even more red and uneven. An Irish tan is not a sunburn; turning red and splotchy is just how some very fair Irish-Americans tan.
I don't buy the schmegegge about Morty sleeping with Moira.
His version of the story was pure schmegegge.
The whole schmegegge was made up to get Liz a little bit of attention.