Um, this might be a little obvious, but have you seen this painting that they keep talking about in the novel? It's a real thing, so click on the link. Go ahead, we'll hang out and wait for you. No, seriously, go look at it first, because it's pretty much impossible to discuss the thing without having seen it.
Now before we launch into the meaning of the thing, a little background info. Dostoevsky saw this painting when traveling in Switzerland, and was so struck by it that he climbed up on a chair in order to look at it more closely. (We have a nice little story about this in his wife's diary, where she's all stressed that security would make them leave.) Okay, so why does Rogozhin have a copy of this painting hanging in his house? And why does he constantly stare at it?
According to Dostoevsky's own notes, and then according to Myshkin and to Ippolit, this painting is pretty much right at the crux of Christian doctrine. Central to the Christian faith is the belief in Jesus's resurrection. But if you look at this image of what Jesus might have looked like after his crucifixion—beat up, bloody, emaciated, laid out on a slab, and really very, very dead (Holbein used a drowned body as a model)—you're forced to come to the conclusion that Jesus was just a mortal guy and that his resurrection is completely physically impossible.
For example, check out Rogozhin's own admission of the effect it's been having on him:
Over the door, however, there was one of strange and rather striking shape; it was six or seven feet in length, and not more than a foot in height. It represented the Saviour just taken from the cross.
The prince glanced at it, but took no further notice. He moved on hastily, as though anxious to get out of the house. But Rogozhin suddenly stopped underneath the picture. […]
"I like looking at that picture," muttered Rogozhin, not noticing, apparently, that the prince had not answered his question.
"That picture! That picture!" cried Myshkin, struck by a sudden idea. "Why, a man's faith might be ruined by looking at that picture!"
"So it is!" said Rogozhin, unexpectedly. (2.4.1-10)
And if that's the case—well, if you're Rogozhin and that's the case, then maybe you decide that your murderous ways are pretty much a-okay, what with no afterlife and all. The novel equates morality with religious belief—and characters who have no faith don't have the slightest moral scruples against doing horrible things. We might not know how every single character feels about religion, but everyone who mentions it definitely goes either one way or the other.
So in a sense, this painting is the kind of like a litmus test. Myshkin looks at it, but then quickly looks away, afraid to make his faith disappear. Rogozhin stares at it for hours and loses his whole moral compass. And Ippolit looks at it and it helps him to glom onto that whole the-universe-is-random-and-doesn't-care-about-you feeling. So, who else should be forced to sit in a small room with this painting for an hour? How do you think Aglaya would react to it? Varya? Nastasya? Is this litmus test not that important for some characters?