Everyone knows that a monarch's crown is never just a fancy, bedazzled hat that looks good with a matching golden wand and throne. It's more than that: it's a visual symbol of power. In this play, the crown is parodied and passed around like a turkey on Thanksgiving.
When York tells us of his plan to kick Henry out, he says, "when I spy advantage, claim the crown" (1.1.253). He gets that the crown isn't a piece of metal with some jewels that a king wears out to fancy parties—it's symbolic of his power and authority. Did you notice how the characters use the word "crown" to mean being king? That's because it's not just about the headwear.
The crown was a seriously important symbol in Elizabethan England. To show a king giving up his crown onstage was pretty dangerous, since monarchs did not like plays that depicted this kind of thing. The thinking went like this: if the audience sees a king give up his crown on stage, they'll start to imagine how the real-life queen might give up hers.
It just goes to show you how powerful the stage was thought to be. If monarchs were afraid to let people even see an actor pretending to give up a crown on stage, then writing history plays back in the day seems like it was pretty serious business.