In one specific circumstance was there a serious problem due to the religion of the Empire. Even then, it was not the religion itself, but the insistence on uniformity. But even with this problem, the Empire continued on for another 6 centuries. This is even considering the various religious conflicts and controversies that embroiled the Byzantine Empire. And, all the while, the Franks were able to use Christianity as a unifying and stabilizing force, as well as a Casus Belli, to strengthen their realm almost up to parity with the East. Its almost as though Christianity was no real threat to the Empire.
Quoting wikipedia verbatim is not much of an argument.
First of all, the East had their own problems. They were not immune to attacks, they had to face the brunt of the initial Gothic invasions, and even after the West fell, there were Ostrogoths within the Eastern borders. They also had to hold the Persian frontier, which was no small task. Putting the entire Empire under one administration does not lessen the threats, it just means that more decisions have to be made by a central authority, which cannot always react to problems on opposite sides of the Empire. That is why they split up the administration in the first place, so that threats in opposite regions of the Empire could be dealt with simultaneously.
This worked just fine for centuries. Even as early as the Antonine Dynasty, Marcus Aurelius managed the day-to-day running of the Empire, while Lucius Verus commanded the forces against the Persians. It took extremely exceptional rulers to be able to handle all the administration of the entire Empire, and those just weren't always available. The only hope was that there wouldn't be major threats during those periods. If there were, it just simply could not be done by one man.
Second, guess what? The *did* send help. Even as late as 468 (just 8 years before the 'end' of the Western Empire), the East and West were trying furiously to retake Africa, because both states knew that Africa was vital to Rome. And when the failed, look, the west fell soon after. Mind you, the Cape Bon expedition to which I am referring, had ships in the hundreds on the combined Roman side, whereas the the previous attempt by the West to retake Africa, that ended in failure, couldn't even muster 50 ships, and their loss doomed the expedition without Eastern help. As it was, it took the East another 70 years to recover enough to be able to effectively threaten Africa again.