To what extent was Christianity responsible for the dark ages of Western Europe?

You are right. This can be a heated question, as is evident from comments, upvotes, downvotes, and votes to close.

TL;DR There is no consensus among historians, so much is sure. There was a crisis in the Western part of the Roman Empire, connected to the collapse of the central authority there, but obvious in demographics, economic and political organization, technology, social development. There is no consensus what caused it, and at all times there were historians (yes, Christian ones, mostly) who disputed that there was a decline.

What decline?

The decline is disputed by some historians, both ancient (notably Orosius) and modern. However, it is manifest in

• demographics (lower population density),
• technology level (no aqueducts any more, no floor heating, no bath houses, no lighthouses)
• economic organization (no more luxury goods and complex production, collapse of the trade network),
• political organization (devolution of authority tribal communities of (Barbarian?) conquerors), and
• social development (why would you want to learn to read if you have neither papyrus to write on nor books besides the bible?).

For more detail on all of these, see Bryan Ward-Perkins' 2006 book "The Fall of Rome and the End of Civilization".

Timeline of disaster

• 196 CE (fall of Emperor Commodus): Military gains power over the choice of the Emperor.
• 235-284 CE: Empire ruled by a succession of Military rulers, most of which are short-lived and assassinated and deposed after a short rule. Diocletian's reforms reorganize the empire into a pre-feudal state with multiple (four) Emperors residing in locations distributed across the empire and therefore able to respond to local rebellions more swiftly. The reforms provide some stability, but the power remains with the warlords.
• 260 CE: Rome abandons the Northern parts (right bank of the Rhine) of the province Germania superior.
• Starting in 381 CE: Persecutions of pagans by Theodosius (especially after 389 CE). This is not the first religious persecution in Roman history, but the ones before targeted small minority groups (Christians). The one under Theodosius was the first large scale one that was essentially run by a minority group (Christians) against the pagan majority (still a majority at the time). Earlier persecutions of pagans by Christians (from 337 CE under Constans II, later under Gratian I and Valentinian II) were relatively moderate, limited to prohibiting certain activities and jobs for pagans and tolerating violence against pagans by Christian mobs (see below).
• 406 CE: Fall of the Rhine limes. Germanic tribes devastate large parts of Gaul and Iberia and establish independent chiefdoms (kingdoms) within the borders of the Empire. The authorities attempt to make arrangements for a peaceful coexistence and to integrate the newly arrived into the military forces of the Empire to repel subsequent waves of incursions.
• 410 CE: Olympius's f**kup leads to the sack of Rome by the Alaric's Visigoths. The Visigoths were at this point not a tribe, but a band of Germanic tribesmen, runaway slaves, and unemployed mercenaries. This was entirely preventable and a result of Roman political infighting. Alaric was the commander of a group of Gothic mercenaries. He was in imperial service, then declared an enemy of the state, and when he then invaded Italy, the Roman side was unable to efficiently conduct negotiations with him or to organize an effective defense. After several years back and forth, he sacked Rome. The Western Roman Empire never recovered, the military organization was practically gone, the Western Roman army after this is more similar to the marauding bands of mercenaries than to the organized force it was in earlier times.
• Early 400s CE: The empire abandons Britannia. There was no enemy force taking Britannia, although it was under attack and asked the emperor for help. He replied that they were on their own. It was not immediately conquered and retained Roman characteristics for the next 1-2 centuries.
• Early 400s - 476 CE: A series of weak Western Roman Emperors, controlled by mercenary leaders who were mostly (but not all) Germanic tribesmen: Stilicho, Ricimer, Odoacer, Aetius, Orestes.
• Mid 400s CE: Rom abandons Pannonia, Noricum, and other provinces.
• 476 CE: Western Rome returns the imperial authority to the Eastern Roman emperor. Eastern Rome does not politically control the Western empire. There are Roman remnant states throughout Western Rome that have not been taken by Germanic conquerors (Soissons, Mauretania). Eastern Rome attempts to conquer the Western parts of the Empire under Justinian, but the attempt largely fails (except for Africa), possibly because of the plague of Justinian.
• 630s CE: The Islamic caliphate conquers most of the Eastern Roman empire (the Levant, subsequently all of Northern Africa, and, still later, most of the Mediterranean islands). They proceed to besiege the Eastern Roman capital, Eastern Rome frequently has to ask the Bulgars for help.

Why did Rome ever fall?

You could argue that Rome fell centuries earlier with the decline of the republic that gave rise to the empire. Disregarding this and focusing on the crisis around the 5th century, there are several possible and probable causes that potentially interacted. For each of those, there are scholars who argue/have argued in favor of it

• Rise of Christianity in the 4th century
• Collapse of political stability in the 3rd century.
• Environmental aspects, changes in climate (cooling in Northern and Western Europe, if I remember correctly)
• Epidemics
• "Barbarian" incursions: These had happened since the beginning of the Empire, but in 406 there was one that the Empire failed to repel in contrast to each one before that.
• Economic decline because of depletion of natural resources
• Economic decline because of a negative foreign trade balance. This lead to a substantial outflow of money in the form of precious metals from the empire (primarily to India) and to successive debas*m*nts of silver and gold coin purity starting with Nero, but continuing whenever the Empire was running low on funds.

It should be noted that there are practically no historical sources detailing the fall of the empire. In an early parallel to today's fake news some scholars were able to cause sufficient dissent about what was happening that scholars and the imperial authorities were apparently unaware of the fact that their empire was collapsing. It is a bit hard to imagine, I admit. And likely some individuals did realize what was going on (perhaps when they were carried off by Alaric into slavery, perhaps when they abandoned their legionary headquarters), but were unable to commit this to the scholarly record of the period.

A particularly illuminating piece of bullshit is 5th century CE historian Orosius's attempt to explain how everything is better in spite of the abundant chaos and decline:

For how does it harm a Christian who is longing for eternal life to be withdrawn from this world at any time or by any means? On the other hand, what gain is it to a pagan who, though living among Christians, is hardened against faith, if he drag out his days a little longer, since he whose conversion is hopeless is destined at last to die? (Orosius, History against the Pagans, book VII.41)

The love of Christ

To assess what role the Christians may have played in the fall of the empire, it is important to understand that these Christians were markedly different from today's mainstream Christianity (advocating brotherly love) and significantly less peaceful than what a look at the new testament and the figure of Jesus Christ might suggest.

Catherine Nixey in "The Darkening Age" makes the argument that they are better described as angry mobs roaming around cities and countryside, burning down pagan temples, murdering pagan priests (or, perhaps, all pagans). At a time when pillaging by Germanic tribes and marauding mercenaries was ripe in the provinces anyway, Christian mobs would not have seemed very remarkable. It is, however, very effective in protecting the members of the mob, reallocating resources to themselves, and gaining a position of political dominance.

This is, of course, a very negative view of the part Christianity played in these times. What is clear is that they emerged from the fall of ancient Rome as the clear and unchallenged mainstream religion in all formerly Roman territories (except those that had been conquered by the Islamic Caliphate or the Tengriist Bulgars), while the pagans vanished from the historical record without a trace. It is conceivable that a systematic genocide of Roman pagans occurred. The later track record of how Christianity dealt with non-believers until relatively recently lends some credibility to this. The fact that Christian historians around this time strictly only discuss Christian martyrs is another red herring.

So is this history or fiction? Did Christians really systematically murder pagan priests and other pagans in order to devastate their social structure, erode their political position, and degrade their cultural and tradition. Of course, it is impossible to prove that. There are other factors that may have helped Christianity become dominant, see this question.