There are many ways one could interpret Ivan Karamazov's parable of The Grand Inquisitor. In fact, it is often regarded as a statement of Dostoevsky's own skepticism of the Christian faith. However, the underlying message appears to be a much more nuanced critique of organized religion rather than the whole of Christianity or religion in and of itself. How does Ivan's parable of "The Grand Inquisitor" speak to the problem of organized or institutional religion? (i.e. complicity, herd mentality, absolute authority, etc.)

"The Grand Inquisitor" is the parable (told in the form of a prose poem) at the heart of chapter five of Fyodor Dostoevsky's The Brothers Karamazov. The parable is related to us, as readers, through a conversation between two brothers, Ivan and Alyosha. Ivan, who's skeptical about the concept of organized religion, is the one reciting the poem to Alyosha, a young monk. Ivan does most of the speaking, with Alyosha breaking in from time to time to ask questions or challenge Ivan's message.
In order to get to your specific question about how the poem addresses the problems of organized religion, let's review the events of the parable itself. In Ivan's poem, Jesus returns to Earth (Seville, specifically) during the Spanish Inquisition. He's immediately arrested, then the Grand Inquisitor pays him a visit. The Grand Inquisitor tells Jesus that he's no longer needed: the Church, in Jesus's absence, and in cooperation with the Devil, has found better ways to meet the needs of people on Earth, and Jesus's own practices no longer align with what the Church wants to do. The Grand Inquisitor explains why the Church is rejecting Jesus by recounting the events of the temptation of Christ, in which Satan presented Jesus with three temptations. Jesus turned down all three, and that's why the Grand Inquisitor doesn't want him to return to Earth.
These three temptations, and Jesus's refusal to be tempted, are key to your question: how does Dostoevsky address the problems of organized religion?
Jesus refused to turn stones into bread.
As the Grand Inquisitor reminds Jesus, Satan urged him to turn stones into bread. Because he can, and because exercising that power would ultimately serve to alleviate hunger among the people:

But seest Thou these stones in this parched and barren wilderness? Turn them into bread, and mankind will run after Thee like a flock of sheep, grateful and obedient, though for ever trembling, lest Thou withdraw Thy hand and deny them Thy bread.

Why did Jesus refuse? Even though he was hungry after fasting, he said "man does not live by bread alone," pointing out that humans need more than food, that the "bread" they need is spiritual as well as material. In the parable, the Church rejects this notion: the Church wants to exert power over the people by providing for them materially. The Church wants the people to follow "like a flock of sheep." 
(This gets a bit confusing, as the parable is a story within a story, but at this point in Dostoevsky's writing, we see skepticism of organized religion. Ivan is a stand-in for the writer.)
Second temptation: Jesus refused to perform a miracle on command. 
Jesus felt no need to show off or to prove his divinity. The Grand Inquisitor questions this and criticizes him for it, saying that people need to see proof of divinity, something miraculous, in order to believe in a higher power. Jesus doesn't agree, and he rejects the idea that he needs to appear better than the people in order to help them or to encourage their faith. 
Again, through Ivan, we see Dostoevsky's critique of the showiness and importance of outward appearance in organized religion.
Third temptation: Jesus refused to grab power.
At the temptation of Christ, Satan offered Jesus control over all the kingdoms in the world. Jesus didn't want it. Bad decision, the Grand Inquisitor thinks: the Church should take as much power as it can possibly get. 
Dostoevsky's analysis of the problems of organized religion, as expressed through the character Ivan, puts the Church's greedy, power-hungry, controlling, appearance-focused profile in sharp contrast with Jesus's humility and goodness. As you mentioned in the original question, the debate here isn't about Christianity specifically. The parable is a comment on the dangers of any organized religion that attempts to dominate people in exchange for security and comfort.
https://www.mtholyoke.edu/acad/intrel/pol116/grand.htm