How can I disable HTML5 content in popular browsers like Firefox and Chrome?

In Firefox you can set media.autoplay.enabled to false in about:config though as of now this may not be sufficient for every case. An extension like NoScript (default settings) could be used to complement the missing functionality or even otherwise used by itself without the about:config setting. In NoScript it's also possible to extend the functionality to whitelisted sites (NoScript Options > Whitelist) via NoScript Options > Embeddings: Apply these restrictions ...

about:config Entries

For Chrome/Chromium, there is HTTP Switchboard (available in Chrome store or github). There is a column in the matrix which can be selectively be turned on or off, which turn on/off HTML5 video/audio/SVG/fonts -- for whatever hostnames you wish.

In Firefox 36.0.4 open about:config and search for media.webm.enabled set this to false and the html5 videos will not play anymore. ;)