Three times. The Third and Tenth Doctors were killed by radiation poisoning in "Planet of the Spiders" and "The End of Time", respectively. The FIfth Doctor died from Spectrox poisoning in "The Caves of Androzani".
Both the Tenth & Eleventh Doctors had close escapes from poisoning. In "The Unicorn and the Wasp", the Tenth Doctor successfully counteracted cyanide poisoning. In "Let's Kill Hitler", River Song used all her remaining regenerations to counteract a poison (which she had herself administered to him) when the Eleventh Doctor was close to death but had not yet lost consciousness.