Kara-Muzra is a member of the RPR Parnas opposition party, which had been led by Kremlin critic Boris Nemtsov who was murdered in February. He also works for the Open Russia movement founded by another opposition figure Mikhail Khodorkovsky.
According to the Guardian, Kara-Muzra's doctors initially suspected poisoning, but later diagnosed him with kidney failure.
"The deputy head doctor of the hospital came out and told us that everything was fine with his heart, his lungs, his stomach. It all had to do with his kidneys. It could have been a spoiled yoghurt or something," Kara-Muzra's father told RBC newspaper.
Although Kara-Murza's father dismissed rumours of a deliberate poisoning, his colleagues expressed fears about the sudden illness, following the killing of Nemstov.
Open Russia project coordinator Maria Baranova told the Guardian she was suspicious and "various activities surrounding public people from Open Russia look strange".
Kara-Murza, who was previously a television and print correspondent and contributor to many well-known Russian and western publications, was also involved in drafting an explosive report on Russia's acknowledged military presence in eastern Ukraine. The report was based on research gathered by Nemtsov before he was shot dead near the Kremlin in February.
The 65-page report, "Putin and the War", claims that Russian troops have been involved in the conflict and that Russia has covertly trained the rebels and supplied them with military equipment and intelligence.