Crystal Clear Picture

If older style computer or TV screens are recorded on film, the scan lines on the picture tubes can become visible, making a mess of the image that was easy to see by everyone but the camera. This is because the TV's scan rate and camera's frames-per-second are not synchronized, and as a result, a black bar repeatedly rolls up or down the screen.

There are two ways to avoid this. First, there are ways a television screen can be specially synced up with the camera. This is surprisingly difficult with older technology.

Second, the images on the TV/computer screen can instead be added as an optical effect in post-production, with the result that now a clear picture appears. This is a less sophisticated effect, so you tend to see it more often only in older productions. Sometimes, the FX-inserted image is suspiciously crisper than everything else around it. Another dead giveaway is the shot suddenly moving to an extreme closeup of the TV screen, with only a dial, button, or paying brand name visible off to one side to even clue the viewer in that they're being shown the same screen that other characters are looking at, and not switching scenes entirely.

Rarely seen on newer shows because of advances in special effects and new monitor technology, but just about any sci-fi series from the 60's or 70's will have at least one example.

Film

 * One of the Master Ninja movies shown on Mystery Science Theater 3000 had the villains watching a surveillance screen that looked suspiciously sharper than the rest of the scene. A similar effect appeared in the Riding With Death movie. (Both movies were made somewhere in the 1970's-early '80's.) The Mystery Science Theater 3000 host segments themselves featured the "Hexfield Viewscreen", a video-phone device that was basically a fancy opening into another room, where actors would stand and converse with the cast. The artifice of this fake screen was further highlighted in an episode where the "image" breaks up with "static"—visibly a scatter of little styrofoam pellets being thrown at the actor from offstage.
 * Near the end of The Three Stooges In Orbit, television executives are watching a demo reel of a new style of Stooges cartoon on a television, when the building gets crashed into. The TV is knocked around like the other props, but the image doesn't, and keeps running even after one would think the set lost power. This, by the way, was not done in a closeup of the screen.

Live-Action TV

 * In the Doctor Who serial The Sea Devils, the Master watches an episode of The Clangers on a "television". The image is inserted using a Green Screen, and in several shots, how this is done becomes painfully obvious: the television itself is just a frame in front of a set of green miniblinds, and "turning the television on" is done by closing the blinds.
 * One episode of The Young Ones didn't even bother with green screen—they simply had a static image taped to the front of the television, and the actors and camera positioned themselves so that the TV was only visible while it was "on".
 * The Goodies made heavy use of this - though considering the 'TV' was 'actually' a super advanced projector thingy, which suspiciously worked exactly like a set of blinds (which were positioned in front of a window, to boot), it was probably a tad less serious.
 * Ridiculously common on MTV reality shows these days. Especially bad when it's used during a shoehorned bit of Product Placement. I honestly have problems gauging how good your new video phone is if the picture is a result of post-production, LG.
 * Amusingly averted in the (presumably low-budget) CBC kids' spy show Spynet. Not only were the villains' computers CRT monitors with visible scanlines, they didn't even have a Viewer-Friendly Interface... just Windows 95.
 * Averted in Gerry Anderson's UFO and partially averted in Space: 1999, where most of the video monitors (even hand-held ones) are real, though the images are in black and white because the production hadn't yet figured out how to synchronize colour monitors to 35mm film cameras. However, the main Moonbase Alpha viewscreen on Space: 1999 used matted-in images, which were therefore in colour.

Western Animation

 * In the animated series Roughnecks: Starship Troopers Chronicles the rolling bar is added deliberately to a background monitor to add 'realism' to a scene.