Not true at all.
For some context, Rasputin was killed by a group of royalists who were worried about the influence that Rasputin had over the Tsarina. WWI was well underway by this point and Tsar Nicholas was spending most of his time at the front (against the wishes of his generals).
Rasputin had worked his way into the good graces of the royal family by claiming to be able to cure the young Alexei, heir to the throne, of his hemophilia. With Nicholas at war, Rasputin was able to install his friends in places of power within the government and wield power beyond what many believed was allowable.
Enter Prince Felix Yusupov, a man who had married into the Romanov family via the Tsars Niece. With the help of two other men, Yusupov came up with a plan to lure Rasputin out and kill him.
The story of Rasputin’s death must be read carefully however, as the most recounted version of events comes from Yusupov’s own memoirs written years later. Historiographically, this puts a lot of what happened that night into doubt, with next to no other primary sources. Rasputin’s autopsy outlines cause of death to be three gunshot wounds, one at close range to the head. There is even a fairly easy to find image of his corpse.
According to Yusupov, prior to being shot, Rasputin was fed cakes and wine laced with cyanide. When he did not appear to be effected by this Yusupov went for the gun. He shot Rasputin in the back, left with his conspirators, and then returned to deal with the body. Upon returning, Rasputin supposedly jumped up and wrestled with his murderers before being shot twice more.
The autopsy found no evidence of poisoning. Furthermore, the poisoner, Dr Stanislaw Yusupov, confesses on his death was of suffering a crisis of conscience and not actually adding the poison.
Now, earlier I mentioned the big H word (historiography) when looking at our key source, Yusupov’s memoir. Why turn Rasputin into this unkillable demon? Well, put yourself in Yusupov’s shoes. He was a member of the royal family and he had, many years ago, shot a holy man in the head because of politics. If you are wanting to sell your book wouldn’t you spice up the telling to make it a bit more interesting, and Rasputin a bit more devilish?
Hopefully this helps. This is my first time responding to a question and I’ve had to quickly dash this off on my phone. Hopefully it is sufficient.
Now, sources for all this include: