None of the main companions have died, unless you count Jack, who has died many times. The only "companions" to die in the new series are Astrid in "Voyage of the Damned" and Adelaide in "The Waters of Mars". Harriet Jones is sort of a companion, but more of a returning character, who died in "The Stolen Earth". Also Emma and Grayla in other media.
If you count Sarah Jane, in universe it has not being confirmed, but in real life, her actor Elisabeth Sladen died of cancer four days before the Impossible Astronaut in 2011.
According to a recent interview with RTD, "Of course Sarah Jane isn't dead, we still haven't shown the last half season." After that, he wants to talk to Moffat and see if they can come up with a worthy way to write her out, or whether it would be better to just leave her fate ambiguous.
If you count River as a companion of the 10th Doctor as well as the 11th, she died in her first appearance.