The light entering the camera from the viewfinder will only affect metering when the mirror is down. When the sensor is exposed, either to take a picture or to enable Live View, the mirror will be up and will block pretty much any light entering the viewfinder from reaching the imaging sensor. This is true of just about any DSLR with a reflex mirror whether it is a Canon, Nikon, Pentax, Sony, etc. (The only interchangeable lens cameras that I am aware of that have an optical viewfinder and no reflex mirror are the Sony SLT designs. For the Sony SLT cameras it probably wouldn't hurt to cover the viewfinder when not using it.)
The only time you need to use the eyepiece cover is when you are using the self timer, shooting a bulb exposure, using a remote trigger, or any other times when you are using automatic metering with the mirror down but not placing your eye to the viewfinder. You don't need to use it in Live View because metering in Live View is done by the image sensor with the mirror up, not by the light meter that is inside the prism housing near the viewfinder.
If you are shooting in a very dark environment with a very strong light behind the camera a little light may leak around the edges enough to bounce around the light box and affect the image, but under normal use it will be much less light reaching the camera sensor than what is reaching the exposure meter through the viewfinder when your eye is up to the viewfinder and you are looking through it.
From page 187 of the EOS 5D Mark III Instruction Manual:
When you use the self timer or shoot bulb exposures and do not look through the viewfinder, stray light entering the viewfinder can cause the image to come out dark. To prevent this, use the eyepiece cover (p. 27) attached to the camera strap.
During Live View shooting and movie shooting, attaching the eyepiece cover is not necessary.
Similar statements are included in pretty much every Canon EOS DSLR manual. Among the ones I have handy at the moment are the 5D II, 7D, and 7DII Instruction Manuals. The concept is the same for any DSLR with a mirror.
The early version of this statement did not include the last sentence clarifying that using the cover is not necessary when using Live View. But in every case I can find, the instruction is included immediately following the instructions on how to take bulb exposures that assume the mirror is down between shots. It is not included in the chapters regarding Live View.