Dorotheos’s name day is June 5, the feast of Dorotheus of Tyre.
Churches that follow the Old (Julian) Calendar keep the same feast on June 18 of the civil calendar. See everything commemorated on June 5.
The saint behind the name

Dorotheus of Tyre
June 5, Hierarch, Δωρόθεος
The Holy Hieromartyr Dorotheus was Bishop of Tyre in Phoenicia. He suffered exile in the great persecution under Diocletian, returned to his flock in the peace of the Church, and shepherded them for many years as a learned and holy elder. In extreme old age, tradition tells, he was seized in the days of Julian the Apostate and died a martyr from the torments inflicted on him.
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.