If the shutter wasn't in the picture, what would you get? Are there any examples of this available? I've searched, but couldn't come up with anything. Is this an impossible thing? Or are the results so unremarkable it's not worth showing? Would it just be a continuous blur of motion?
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.
I think it would be unremarkable as to be not worth showing.
Knowing that motion pictures are just a regular sequence of still images, what separates frames is, well, holding the image frame still for the exposure time (which is somewhat controllable, but generally about half of the frame rate period). Yes, the shutter does blank the light from exposing the film or sensor for a brief period of time (or in the case of electronic shutter, blanking and resetting the sensor).
The excellent YouTube video How a Film Projector Works (only 9 minutes) by Bill Hammack, a.k.a. engineerguy, shows how a movie projector operates. I can't recommend this video enough (as well as all of his videos). There is so much information presented, incredibly clearly demonstrated.
Early in the video, note that the projector does not smoothly move the film strip past the lamp and lens. If that were the case, it would be a blur of information, and you could barely make out some of the film detail. Instead, the film frame is held stationary while its image is projected. Then, when the film is advanced, a rotating shutter blocks the lamp from projecting the image through the lens. I like to make the analogy of a stage play closing the curtains while the stage set is changed, then the curtains are drawn to display the new set. You didn't see the stagehands and props moving. You only get to see the placed sets.
The process of capturing film is almost exactly the same, only rather than using a lamp to project through a developed frame, light coming through the lens has to expose a stationary frame of film for a fixed period of time. In the projector video, note that a single frame is actually projected 3 times by use of a 3-bladed shutter wheel, in order to increase the flicker rate (apparent frame rate) beyond the flicker fusion threshold of about 60-70 flickers per second.
But in a film video camera, a single shutter wheel exposes the stationary image frame for (about) half of the single frame period. This is controllable by a sort-of variable-sized rotary disc shutter wheel. This control over the single-frame exposure time is known as the shutter angle. The reason to control the shutter angle is to allow for more or less motion blur to be recorded in the film frames.