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.
The first thing Warwick does when he captures Edward is take off his crown and declare, "Henry now shall wear the English crown" (4.3.51). And when the gamekeepers find Henry in the woods, they ask him where his crown is. Obviously a real king would never travel without it. Henry replies:
My crown is in my heart, not on my head;
Not decked with diamonds and Indian stones,
Nor to be see. My crown is called content;
A crown it is that seldom kings enjoy. (3.1.62-65)
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. Tellingly, Henry doesn't have a crown on... because he doesn't have any power.
Henry's wife Margaret knows that the crown is worth more than its weight in gold, too. When she captures York, she puts a paper crown on his head, and mocks him: "Ay, marry, sir, now looks he like a king. / Ay, this is he that took King Henry's chair" (1.4.97-98). Her fake coronation is done to ridicule York for wanting to be king. But it also points out something deeper to the audience.
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.