I see this all the time: a character is doing something on a computer or watching TV. Instead of filming the actual content of the screen, they film the scene with a fixed camera and add a layer with the content they want to show during post production.
Here is an example (from a TV series named Underemployed, Season 1 episode 1 at 18:42).
In this case, you can see that the perpective correction is wrong (the bottom right corner doesn't match the actual corner of the screen), and I'm really wondering what I'm looking at. It looks like a deformed iPad application displayed on a cheap PC, I really don't know.
What kind of problems occur when you film an LCD ? Is it a technical issue, or is there another problem I'm not thinking of ? Maybe a logistic problem (like.. the actor doesn't know how to use a computer, or they should synchronize the actors play with the content played on the TV..).