Russia's a huge country, so results will vary.

Both the Revolution and World War I were accompanied by large-scale defection and disillusionment from the Orthodox Church, especially among workers and soldiers, although peasants tended to remain religious. Naturally, the Church was quick to side with the Whites. Strictly speaking, religion was never "abolished" but it was discouraged as Tsarist relic. During the early 1920's, there was a halfhearted attempt to institute 10-day workdays(so they could get rid of those pesky Godly Sundays). Other religions were differently affected; IIRC the Bolshevik attitudes towards Islam was much more focused on sympathy for these "oppressed minorities" than abolishing their religion.

Describing the aftermath of the battle of Moscow, John Reed, an American Communist, wrote:

The Holy Orthodox Church had withdrawn the light of its countenance from Moscow, the nest of irreverent vipers who had bombarded the Kremlin. Dark and silent and cold were the churches; the priests had disappeared. There were no popes to officiate at the Red Burial, there had been no sacrament for the dead, nor were any prayers to be said over the grave of the blasphemers. Tikhon, Metropolitan of Moscow, was soon to excommunicate the Soviets….

Also the shops were closed, and the propertied classes stayed at home—but for other reasons. This was the Day of the People, the rumour of whose coming was thunderous as surf….

Already through the Iberian Gate a human river was flowing, and the vast Red Square was spotted with people, thousands of them. I remarked that as the throng passed the Iberian Chapel, where always before the passerby had crossed himself, they did not seem to notice it….

Sauce: John Reed, Ten Days that Shook the World

Ninja Edit: That said, the godless Renaissance was mostly over by the end of the twenties. The Party began to reconcile itself to the Church, and Stalin conveniently replaced the Tsar as God-on-Earth. By the time of the Great Patriotic War, the Orthodox Church was a major asset for the Stalin government.

The revolutionary period was an intense period of upheaval within the Russian Orthodox Church, so it's not just a matter of Orthodoxy vs. Bolshevism. There were important debates going on within Orthodoxy (and among Christian reformists who were distancing themselves from the institutional church). The intelligentsia at the time was overwhelmingly atheistic, in part because they tended to see Orthodoxy as an inherently conservative force. The ROC had, since the time of Peter the Great, been turned into basically a bureaucratic department of the state, and so the revolutionary-minded intelligentsia naturally directed part of their critique at Orthodoxy. And to be clear, there were definitely forces in the ROC who were rather satisfied with the arrangement and who defended the collusion of Orthodoxy and autocracy--much of this sentiment stemmed from a broader anti-Western attitude, since political reform at the time was often expressed in terms of a "Westernization" that was thought to be destructive to tradition Russian and Orthodox values and ways of life.

At the same time, there were definitely some serious reform movements happening in the ROC at the time. In the first half of the 19th century, the early Slavophile movement had already begun to challenge the church's subordination to the state (see Walicki, The Slavophile Controversy; Rabow-Edling, Slavophile Thought and the Politics of Cultural Nationalism), and in the latter part of the century, the great Vladimir Soloviev, Russia's first truly modern philosopher, had pushed that critique quite far. The momentum picked up so heavily that by the start of the 20th century, there was a serious push for ecclesiastical reform that culminated in the All-Russian Sobor of 1917-18, which, sadly, happened to coincide with the revolutions of the same time. The Sobor ("council") reinstated the Moscow Patriarchate, which had been abolished during the Petrine reforms, and instituted several other reforms that were intended to move the ROC in a more progressive direction. For more on the Sobor, see Destivelle, The Moscow Council (1917-1918), just released.

The Sobor was actually a fairly moderate expression reformism within the ROC. The search for freedom for the Church had long been associated with the desire to get the Church behind politically progressive causes: the Slavophiles had argued for the abolition of serfdom (which they as symptom of spiritual sickness in Russia), Soloviev had been an ardent defender of freedom of conscience and freedom of the press and a critic of all forms of religious coercion, and by the start of the 20th century, the "Russian idealist" movement had begun pushing for serious political form along liberal and/or socialist lines (there were prominent Orthodox voices in the liberal Kadet party in the Duma, for instance). Some Orthodox clergy even bordered on outright revolutionism, e.g., the priests who formed the Christian Brotherhood of Struggle. Here's the thing that may be shocking: a fair number of Orthodox clergy actually had a positive view of the communists at first, some emphatically so, and others with some reservation but a general hope that they could ultimately be a force for good with a little Christian influence (see Roslof, Red Priests). In general, even those who were rather strong critics of the revolutionary intelligentsia often tended to be strong advocates of some kind of serious social and political reform (see Hedda, His Kingdom Come).

Then the Bolsheviks had to go and mess everything up. Roslof notes that Lenin's pre-revolutionary writings place high value on the freedom of conscience and make no mention combating religion through the use of coercion or violence--he didn't want to create an atheist version of the autocracy, basically. That's in keeping with what Marx's idea of religion as "opiate" actually means: religion is a painkiller, a coping mechanism, so when you get rid of the underlying disease, you get rid of the need for religion. Lenin and some of the other revolutionaries seemed to think initially that once the revolution was successful, religion would rather quickly and easily fade away on its own.

Pretty quickly, though, they started attacking the ROC and Orthodoxy directly. For one, the Soviets invalidated the decisions of the All-Russian Sobor, except for the reinstitution of the patriarchate, which they then attempted to control. In 1922, the they put an unruly Patriarch Tikhon on house arrest and then took advantage of internal conflict within the ROC to encourage a second All-Russian Sobor, led by the most radical wing of the clergy, that realigned the Church with the Soviet cause. This led to a schism between the mainstream, modern Church and this radical wing, the so-called Living Church, which for a brief time was recognized as the "official" Church in Russia. However, the latter half of the 1920s, Tikhon's successor, Patriarch Sergius, made efforts to reconcile the canonical ROC with the Soviet government, and 1927, while in prison, he pledged the support of the ROC to the Soviet regime, most likely in hopes that it would help to ease up Soviet persecution. While that crippled the Living Church experiment, it did solidify a new schism within Russian Orthodoxy, leading the church-in-exile--the Russian Orthodox Church Outside of Russia (ROCOR)--to break ties with the Moscow Patriarchate.

Anything beyond that is outside of my competence. I mainly focus on Orthodoxy leading up to 1917.

Just to add to /u/trampabroad's comment, it's worth noting that religion in Russia was far less institutionalised than those in Western Europe (the Catholic Church being the obvious counterpoint). This was most obvious in the status of its priests who, at village level, were generally barely literate poor peasants who tended to supplement their impoverished existence by charging for most rites/functions. As a result, there was very little respect for the Church itself amongst the peasantry. As with everything else, the peasants put the land question above all else.

Hence there was little reaction to the dissolution of the Church post-revolution: its wealth was seized by the state, its lands distributed to peasants and the communes often began electing their own priests. All of which served to weaken the official hierarchy to the point of irrelevance.

Very different was the popular expression of religion. Despite their attitudes towards Church institutions, the Russian peasantry as a body remained intensely devout. Religious ceremonies/rituals (not just christenings, wedding and funerals but also the myriad saint days/festivals) continued to shape the rural calendar. And, of course, no peasant hut would be complete without the corner shrine, replete with holy icons, to the local saint.

The Bolsheviks completely failed to break down this base level of religion (which went much deeper than just religious forms). If anything, the opposite occurred - with peasants often interpreting the new Marxist regime via the prism of old religious customs. There were many stories of peasant households simply including portraits of Lenin/Stalin in their icon corner, amongst the saints.

By the late 1920s the most serious efforts at undermining popular religion - civil rituals (for baptism, marriage and funerals), the League of the Godless, atheist processions on religious holidays, disavowal of traditional saints' names, etc, etc - had largely petered out. This opened the door to the rehabilitation of the Church in the 1930s, with the constitution of 1936 (which relaxed hostility towards priests and their families) as semi-official recognition of this.

And finally, with regards the original question, it's worth pointing out that the perception of the peasantry as loyal to alter and throne was just that - a perception. The Russian peasantry proved to be far less pliant and accepting of the existing social hierarchy than pre-revolution platitudes suggested. Peasants had no problem jettisoning the Tsar or the Church when they stood in the way of land reform.

Sources

I didn't draw too much on it above but Moshe Lewin has an excellent essay on popular religion in his Making of the Soviet System. For religion under the NEP and Stalin I'd recommend, respectively, Richard Stites' essay in Russia in the Era of NEP and Sarah Davies' Popular Opinion in Stalin's Russia.