It is not possible to do this because of the browser’s security model. If it was possible, that would be a security problem and would have to be fixed.
Although letting the embedding site know the height of a third party webpage when embedded in the page seems harmless, this can leak information to the embedding site that the browser’s user wants to keep private. For example, http://www.facebook.com/ renders differently depending on whether or not you are logged in, so if my website can work out the height of <iframe src="http://www.facebook.com/"> then I can work out whether or not you are a facebook user, something you probably don’t want me to know.
The information leakage would be similar to the infamous CSS History Leak in that it would reveal information about the user’s relationship with the third-party site just by “linking” to that site (in this case with an iframe instead of a link). Browser vendors had to plug the CSS History Leak, so I suspect if you could work out the height of a third party site rendered in an iframe in any browser, the vendor would have to fix that too.
The information leaked would be anything that can be inferred from the height of a page when rendered for a user using their cookies (which the browser will send even though rendering in an iframe inside a different domain’s page). The specific risks depend entirely on the nature of the embedded site being “attacked”. E.g. I could get an idea of how much stackoverflow activity someone visiting my site has by getting the height of https://stackoverflow.com/reputation which is different for different users.