According to Harry, yes. But we don't know.
I'm going to start by disagreeing with Toomai's answer. Barty Crouch Junior, under the guise of Mad-Eye Moody did claim that he had Dumbledore's blessing for showing them the Unforgivable Curses. That is not to say that Dumbledore was cool with it. I'm not prepared to take the word of a crook as given. Bearing in mind Crouch's allegiance was not to Dumbledore, I don't think he'd be above lying to cover his tracks. If a teacher says Dumbledore's fine with something then the kids are hardly going to doubt him. It's a very easy lie to pull off - if it was a lie.
So what did Dumbledore know about lessons, and the school in general? Harry certainly thought Dumbledore was omniscient about Hogwarts.
"He's a funny man, Dumbledore. I think he sort of wanted to give me a chance. I think he knows more or less everything that goes on here, you know."
(Philosopher's Stone, Chapter 17, The Man With Two Faces).
And Dumbledore himself says that he's been keeping a close eye on Harry (and, by extension, his lessons).
"I defy anyone who has watched you as I have - and I have watched you more closely than you can have imagined - not to want to save you more pain than you had already suffered."
(Order of the Phoenix, Chapter 37, The Lost Prophecy).
How Dumbledore did his watching we can't be sure. It's likely that he communed frequently with the teachers but he may have spied on Harry in some other ways.
But there were plenty of things that went on at Hogwarts of which Dumbledore was utterly oblivious. For instance, James, Sirius and Peter became Animagi under his very nose without his knowledge.
"But I sometimes felt guilty about betraying Dumbledore's trust, of course...he had admitted me to Hogwarts when no other Headmaster would have done so, and he had no idea I was breaking the rules he had set down for my own and others' safety. He never knew I had led three fellow students into becoming Animagi illegally."
(Prisoner of Azkaban, Chapter 18, Moony, Wormtail, Padfoot and Prongs).
Dumbledore also had no idea about the Room of Requirement (which he unwittingly used as a toilet), Rita Skeeter's methods for sneaking into Hogwarts, the use of Polyjuice Potion by the trio in Chamber of Secrets, Draco Malfoy's plan to use Vanishing Cabinets and - of course - the fact that Mad-Eye Moody was being impersonated by Crouch all along.
So, whilst Harry thought Dumbledore knew everything that happened at Hogwarts, there were clearly things he didn't know. We can't say definitively whether Dumbledore knew about Crouch-Moody's lessons or not. However, I would err on the side of guessing that he didn't. There are a few factors behind why.
- Dumbledore trusted Moody. He had no idea that Moody was not to be trusted until the very end of Goblet of Fire. Up until that point, Dumbledore would have had no reason to be suspicious of Moody's lessons than anybody else. He trusted Moody enough to appoint him as a teacher. It would've been strange if he then immediately begun to investigate his lessons for wrongdoing.
- Dumbledore doesn't seem to take a great interest in the curriculum. Teachers in the series are always shown to have pretty-much the final say in what they teach and what they don't. Dumbledore is never shown interfering with lessons.
- Dumbledore doesn't really speak to the students. Student feedback at Hogwarts appears to be approximately nothing. Dumbledore is never shown asking the students whether their lessons are any good or not. (They frequently aren't: see Lockhart and Trelawney).
- Learning about Dark Spells is OK. The passage Toomai quotes appears to show that learning about Dark Magic is part of the curriculum (at least for older students): "Now, according to the Ministry of Magic, I'm supposed to teach you countercurses and leave it at that. I'm not supposed to show you what illegal Dark curses look like until you're in the sixth year." Even the Ministry of Magic is fine with showing students what Dark Magic looks like. As Moody says, if you don't know about the spells, how can you defend yourself against them?
- Dumbledore wouldn't have stood for it. I've argued elsewhere that Dumbledore himself performed Unforgivables. But only when the situation demanded it. I suspect he would've been fine with Moody demonstrating the Curses on spiders. But performing them on students as Moody did was illegal and unnecessary. If Dumbledore had known what was going on he would have put a stop to it.
