When you burn a section of the Capital, I'm assuming you can no longer play cards there. Otherwise you could just develop it again. If you burn the battlefield, can an opponent still play Units and attack you?

If one side of capital is burning You can still play cards there and use developments but they will not give You any hit point - because it's already burning. So If you put the battlefield on fire - nothing happens! Burning side don't mean that it's destroy just overrun by enemy's. You can use them later too.

Awesome, thanks for the clarification.