Russia's entrance into WW1 had little to do with the Russian Revolution. How Nicholas responded to domestic politics had everything to do with the Revolution. And that was influenced by Rasputin, right up until he was assasinated in 1916 by members of the Royal family.
Well it was something Rasputin was very much opposed to and yet it happened. Such was his power, or lack of it. The war was a huge trigger for the train of events which resulted in February 1917. Not least the major losses, the food shortages, the loss of confidence in the regime etc etc. Not really controversial, given similar circumstances had helped produce the revolution of 1905 too. But I'm sure one could find some historians who wouldn't consider the social stresses important.
Making Rasputin some sort of evil mastermind which then led to the better part of a century of official atheism is the stuff of legends. What need is there for Rasputin when 'normal' society produces a man like Pobedonostsev, who really did have power and influence to help shape the context of a repressive society which led to revolution? The Orthodox church has a long history of state sanctioned repression of other religions. "Inciting religious discord" (the thing which the Russian courts have been using to class JWs as 'extremist') has an equally long history of being used against minority groups. It's not just JWs caught up in this, it's Baptists, it's Muslims etc. etc.
A good piece on the broader context and why this is worrying.