What would a video look like without the shutter separating frames?
The answer to this will differ based on if you are really talking video or film.

In either case, you'd have no control over shutter speed, which would mean you were locked at the exposure time of the frame rate you were shooting in.  This would greatly limit your exposure options which would either impact ISO/film grain or the depth of field or both.

With no shutter on film, the film would be exposing as it advanced and would thus be a streaked mess.  It would be stopped for a longer time than it's moving in most cameras, but you'd still be able to see the streaks distorting the image.

With no shutter on video, you fair a lot better.  Since the sensor itself isn't moving, you'd just have continuous scanning that wouldn't be that noticeably different.  Practically speaking, you still have an electronic shutter though because there is a blanking interval where the sensor has to read and dump charges, so there is going to be down time when the sensor isn't collecting photons and during this time, the "shutter" is effectively closed.  In fact, it's fairly common for video cameras to use a pure electronic shutter rather than a mechanical one.