That Rasputin is too-well known in the public mind seems beyond reasonable doubt, though it follows along with the (rather unseemly, if you'll allow a touch of editorial) warm feelings and occasional hagiography for the Romanovs; it would be perfectly possible to explain the end of Tsarism without ever mentioning his name. At the end of the day, the Romanovs were holding the pag when the war was lost, and it's implausible that in temporal proximity any man not the Tsar himself would have been able to deal with the structural problems in Russia.
It's problematic for any Rasputin conspiracy theory that, before the war, he never appeared interested in policy - for example, foreign policy, where his views are characterized by Dominic Lieven as without influence, with his commentary being "few and Delphic," hardly the behavior of a mole or demon. His use of his bizarre relationship with the royal family appears to be mostly concerned with grift, using his pre-and-early-war influence to obtain government positions for his hangers-on. The most prominent of the few direct influences he is alleged to have on government is the dismissal of Vladimir Kokovtsov, prime minister 11 - Jan 14, which Robert Massie attributes to the mutual dislike between Kokovtsov and Rasputin, dating from a meeting in Feb 12. Conversely, Lieven attributes the dismissal to the Tsar lashing out at the cabinet, combined with government controversy over Kokovtsov expansion of the state liquor monopoly for funding.
Unfortunately, it will always be impossible to nail down the real degree of Rasputin's influence from the nature of his position, which can only be reported by historical documents in personal and highly biased terms - especially given the channel of power that he would have wielded - the Empress.
It's nearly impossible to under-emphasize how this plays into every depiction or analysis of Rasputin. Nicholas himself appeared somewhat affectionate towards Rasputin, but the vast majority of the conspiracy that swirled around him in the press (and the alleged channel for the aforementioned dismissal of Kokovtsov) pertains to his relationship with the Empress Alexandra. Despite his influence over policy being tenuous at best, there's no question that his ability to comfort the haemophilic Tsarevic Alexei made him critical to Alexandra's life. Thus developed an image in which Rasputin was a cancer at the heart of Russia, cuckolding the Emperor and sleeping with the Empress and her daughters. This played into a longstanding criticism of the monarchy in which Alexandra - born Alix, daughter of Louis IV of Hesse-Darmstadt, making her the granddaughter of Queen Victoria. It would be hard to find a worse combination, considering that ever after Napoleon either England or Germany was thought to be the principal threat to Russian imperium.
Dislikes of Rasputin and the Empress quickly became the same phenomena, especially after the start of the war, exemplified in the words of one of his assassins:
"The Tsar's ministers who have been turned into marionettes, marionettes whose threads have been taken firmly in hand by Rasputin and the Empress Alexandra Fyodorovna—the evil genius of Russia and the Tsarina... who has remained a German on the Russian throne and alien to the country and its people."
Regardless, the Swiss tutor to the royal family observed that as Alexandra began to actively seek Rasputin's advice, it "simply confirmed her 'secret wishes'".
Whether or not Rasputin had been present in the court, it's far-fetched that similar trends would not have developed. Alexandra was widely thought a foreigner, her husband with few achievements to his name, lacking personal ability or willpower, especially compared to his martyr grandfather, in a prewar Russia filled with social strife. Clearly there was much discontent focused on Rasputin, but what would have been the difference, a few months at most? Sexual degeneracy is a common conspiracy theory, and unlikely to be rooted in any real evidence, as opposed to being an after-the-fact explanation at the time for why Russia was losing the war.
Summary: Rasputin's influence was less over the emperor than the Empress, which played into separate misogynistic and xenophobic beliefs about her that led to his prominence. Despite this, there is little evidence beyond hearsay in the context of these popular fantasies that he meaningfully altered the course of the war.
The End of Tsarist Russia: The March to World War I and Revolution. Dominic Lieven, Penguin, 2016.
George, Nicholas, and Wilhelm: Three Royal Cousins and the Road to World War I. Miranda Carter, Knopf, 2010.
Nicholas and Alexandra. Robert K. Massie, Random House, 1995.