Canon says turn it off on a tripod. Some lenses respond differently, and it depends on the focal length and conditions. For example, using the 500mm F4 with 1.4xII extender, you can get shake even on a tripod with cable release, just from the wind. I have observed IS steadying wind shake.
Some IS lenses do seem to induce motion when on a tripod. The 100-400 is almost spooky in this respect.
Many IS lenses--the 28-135 comes to mind--cause serious motion blur with exposures of 1-5 seconds on a tripod. Turn 'em off.
Battery life. Although Canon's manuals are silent on this, Sigma clearly states reduced battery life as a side effect of OS (their version of IS). Just listen to the IS motor and gyros on the 500mm, and you'll know the battery must be draining.
Turn it off on a tripod. Otherwise on.
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Bill
40D circa 2007, 20D circa 2004, Rebel XT circa 2005, Nikon F circa 1967