i) births and deaths
31 March 1930: birth of Michael Hart, who directed The Space Pirates (Second Doctor, 1969).
ii) broadcast anniversaries
31 March 1973: broadcast of sixth episode of Frontier in Space; last appearance of Roger Delgado as the Master. It turns out that the Daleks are behind it all; the Doctor is injured but he and Jo go to track down the Dalek army.
This is actually the first episode of Doctor Who that I clearly remember seeing when originally broadcast, a couple of weeks before my sixth birthday. I didn't know what was going on, but when the sinister man in the beard said "I've brought some old friends along to meet you", I knew this was something good.
31 March 2007: broadcast of Smith and Jones, starting Series Three of New Who; first appearance of Freeman Agyeman as Martha Jones. Martha's hospital is abruptly transported to the Moon, where the Judoon are tracking down a criminal.
iii) date partially specified almost in-universe
The opening scenes of the second Cushing film, Daleks' Invasion Earth 2150 A.D., take place on 31 March; at least that's what Tom Campbell, played by Bernard Cribbins, thinks the date is when he wakes up after being bashed on the head.