If I was running a http proxy, would it be okay to insert a banner at the top of each page (like this) to fund the server?
I'm not a lawyer, but common sense tells me the answer is 'no' - it would not be okay to earn money off of someone else's work without their permission.
It doesn't apply to your example, because there's no commercial intent. But if there was a banner ad there...
The law goes further than that [not a layer]. Every post is the copyright of the poster if reproduced in its entirety. So if the rights holder (or by proxy, the forum owner) requests that they do not reproduce their content then they would have to abide (ignoring they're in CN and would ignore it). Commercial use / monetisation doesn't matter in this case, if there is no permission or license then they cant reproduce content in its entirety.
[As an aside, it gets more interesting in a dispute between the poster and the forum operator.]