I am not sure if this is the right Stack Exchange website, but anyways: If a flash game was to have a copyright notice, would it be legal for me to post it on my website? This wouldn't be for any monetary gain, as I would not be running ads over their games.
Unless you have permission, you can only publish works of others which aren't in the public domain under certain conditions. You may quote fragments of works; you may use it in satire; there's a "fair use" principle; and there are a few other conditions.
Posting a game verbatim on your website doesn't even remotely qualify for any of those conditions.