So…what exactly is the Crystal Palace? In 1851, The Great Exhibition was held in London. It was basically like a World's Fair, where everyone got together to show how cool and modern things were those days, and especially how much cooler and more modern England was than France. Architect Joseph Paxton said something like, "Hey, let's build a building made out of glass and iron, but mostly glass." The Crystal Palace became a symbol of modernity and technology. When Nikolai Chernyshevsky wrote his radical socialist novel What Is to Be Done?, he used the Crystal Palace as a metaphor. If we all become socialist, he said, we can turn society into a Crystal Palace. And then we can do anything we want!
And that's where Dostoevsky comes in. The Underground Man first brings up the notion of the Crystal Palace in Part I, Chapter Seven when he's talking about how awful things will be if we ever figure out all the rules of nature (because then man won't have free will anymore). It is at this moment, he says, that the Crystal Palace will finally be built. Then he digresses for a bit. When he comes back to it, he's got a few reasons why we shouldn't go along with Chernyshevsky.
The Underground Man talks for a bit about the dangers of establishing an ideal: once we do, we'll never settle for anything less. This is part of the danger with wanting or expecting a Crystal Palace of utopian perfection; we'll spend our whole lives being dissatisfied if we never get it.
The Underground Man knows that "in the Crystal Palace, it [suffering] is unthinkable." As he's told us time and time again, suffering is man's best friend. We enjoy it – inflicting pain is our way of proving that we have free will and aren't beholden to the rational laws of nature. Living in a Crystal Palace would mean giving up suffering, which would mean abandoning free will.
More than anything, the Underground Man rejects the idea that he can't "stick [his] tongue out at" the Crystal Palace. This is highly problematic for a man who makes it his life's work to mock everything.