@Mookey's solution worked for me, but I would like to expand on it a bit more.
I too had the same problem on Ubuntu 15.10 with the open source radeon drivers. Video playback caused severe tearing during high-motion scenes. One video caused lots of tearing in VLC but next to none in Totem.
The xorg options for radeon point at the solution:
XV_VSYNC ... If you are using a compositer, this does not prevent tearing because Xv is rendering to an offscreen buffer and the compositor copied it to the front buffer.
If you install CompizConfig, you can adjust Unity's Composite plugin—among a great many other things. Search for it in the filter dialog box. If you mouse-over the "Unredirect Fullscreen Windows" option, the tool-tip reads:
Allow drawing of fullscreen windows to not be directed to offscreen pixmaps.
Turning the option off (it defaults to on) will use the offscreen pixmaps, allowing the radeon driver to do its work. If this works, the effect will be nearly immediate. No need to close CCSM or reboot. Just restart VLC or the offending application and try it out. If it doesn't work, reset the option to default.
It is somewhat ironic that a feature introduced to prevent tearing actually causes it. But the feature was introduced to fix performance issues with the compositer. If you disable the option, you may have performance issues with full-screen games.
So instead of disabling it entirely, you might try editing the "Unredirect Match" option to remove the applications which are giving you trouble. Oddly, the entry for VLC is wrong. It is listed as
xprop lists its class as
vlc, all lower-case. Perhaps this is a bug. When I fix the spelling, but leave Unredirect on, VLC no longer tears.
Ultimately, we need a replacement for X, such as Wayland or Mir, which will finally make tearing a relic of the past. Tearing is a common issue on many linux installs, and it's very difficult to stamp out.