Will lens softness be apparent on DSLR video footage? The one-line question above had to be vague, so let me elaborate:

Let's say that I'm using a Canon T4i (doesn't necessarily have to be that camera) with a Canon 50mm f/1.4 USM lens to shoot 1080p video. I know that this particular Canon lens is naturally not very sharp at 1.4, and this would probably be apparent on a full-size photo when blown up to 100% scale on a screen.

However, it seems reasonable to me that, even if the lens is soft, if I were to shoot video at f/1.4 and then view it at 100% scale on a screen, the video is still so small in resolution that the lack of sharpness may not be very visible, similar to taking the photo previously described and scaling it down to 50%. 

In this way, I want to see if I can get away with using this lens at 1.4 and get results that look very good and really sharp, despite the lens's apparent natural limitations (and, of course, assuming proper focus on my part).

Does my theory have any truth behind it?
Your theory is correct, video is much less demanding of resolution than stills (1080p HD is only 2 megapixels). However you'll never get a truly sharp video image out of a Canon DSLR due to the strange interpolation/line-skipping that is used to produce a video frame from the full image sensor resolution, the result is always a lot softer than you would get from a dedicated HD video camera (or strangely the much cheaper m43 Panasonic GH2).