Katarina sacrifices herself to save the Doctor in The Daleks' Master Plan. If Sara Kingdom is counted as a companion, she was killed by the Time Destructor in the same serial, which would also count as the Doctor just not being able to save her.
Peri possibly dies and is resurrected by the timeline being changed in Mindwarp; alternatively, her "death" is a trick played on the Doctor by the Valeyard and never happens. (See also Peri and the Piscon Paradox.)
The Doctor kills Kamelion, at its request, in Planet of Fire.
Jack Harkness is killed by Daleks and resurrected by Rose/Bad Wolf in The Parting of the Ways.
River Song sacrifices herself to save the Doctor (an earlier incarnation to one she travelled with) in Forest of the Dead.
(As per Daniel Roseman, if Astrid Peth counts as a companion, (the Doctor offers but she never makes it on to the TARDIS), she sacrifices herself to save the Doctor in Voyage of the Damned. Also, Adelaide Brooke does travel once in the TARDIS, and then kills herself to correct the timeline in The Waters of Mars.)
Book companion Roz Forrester dies in battle in the book So Vile a Sin.
Audio companions Lucie Miller and Tamsin Drew die while defeating Daleks in To The Death.