Okay, so this isn't technically a "fan" film--it's a slickly produced BBC-TV parody film starring Rowan Atkinson, Jonathan Pryce, Hugh Grant, Joanna Lumley, Julia Salwaha, and Richard E. Grant, made for Comic Relief back in 1999.
It's actually sort of non-canon canon. The man who wrote it, Stephen Moffat, eventually went on to become head writer for the Doctor Who series proper.
Behold--"The Curse of the Fatal Death."
For further information, the Wiki article's got it all: