2nd September 1911: birth of Eileen Way who played the Old Mother in An Unearthly Child (1963), the old woman in the woods in Daleks - Invasion Earth 2150 AD (1966) and Karela in The Creature from the Pit (1979).
2nd September 1994: death, two days after his 60th birthday, of Roy Castle, who played Ian in Doctor Who and the Daleks (1965), the first of the Peter Cushing films.
ii) broadcast anniversaries
2nd September 1967: broadcast of the first episode of Tomb of the Cybermen, starting Season 5. The Doctor, Jamie, and new companion Victoria land on a deserted planet and encounter an archaeological expedition exploring the eponymous tombs. But they may not be as dead as all that...
2nd September 1978: broadcast of the first episode of The Ribos Operation, starting Season 16 (the Quest for the Key to Time). The White Guardiuan visits the Doctor and gives him both a quest - the Key to Time - and a new companion, Romana. Landing on Ribos, the two Time Lords are trapped with the savage shrivenzale....
(Incidentally, I love that the four episodes of these two stories, iconic in very different ways, were broadcast on exactly the same calendar dates eleven years apart.)
2nd September 1995: release of Downtime - I wouldn't normally note the release of spinoff video like this, but the reunion of Victoria, the Brigadier, Sarah Jane Smith and the Yeti is quite remarkable. Usually in a good way.
2 September 2011: first broadcast of The Gathering (Torchwood). Two months on, the world has become chaos and there's something about Shanghai and Buenos Aires being antipodes? Gwen and Jack get close.
iii) date specified in canon
2nd September 1666: The Fifth Doctor starts the Great Fire of London, as shown in The Visitation (1982); I guess we assume that most of the 17th-century scenes in the story are set on that day.