Thankfully LCD/LED screens are much easier to photograph than old CRT screens, but there are still a good number of things that can go wrong:

• Glare - this is light reflecting on the monitor and obscuring the image, move the camera around so you don't see the reflection or place something between the monitor and light source to block the light.

• Image too dark/bright - the camera will assume you are photographing something that has middle brightness, a mostly white monitor will be under exposed and a mostly black monitor will be over exposed, with a "real" camera use exposure compensation (or manual mode) to get the brightness just right, with an iPhone tap the screen to focus on a part of the screen that is middle brightness (middle gray works best).

• Vertical/horizontal stripes - those are caused by flicker in the florescent tube on the back of the LCD screen (theoretically shouldn't happen on LED screens, but I don't have one here to test it), you have to set the camera's shutter speed so it catches an entire cycle (if you're power is 50Hz 1/50 or 1/25 will do, if your power is 60Hz 1/60 or 1/30 - USA is 60, most of Europe is 50) - I have no idea how to do this with an iPhone

• Round stripes - this is moire pattern, it is caused by the camera behind fooled by the "grid lines" between pixels - move in closer, move out farther or change angle until they disappear.

• Random dots - this is noise, any acceptable camera (including the iPhone) will only produce visible noise in low light conditions and a screen is pretty bright so this shouldn't be an issue - but it can happen outside the screen or in dark areas of the screen - you can try to play with the screen brightness or add more light (add light for off-screen noise only, and be careful about glare) - in this case it's probably best to remove noise in post-processing

• Strange color casts - this can be caused by the fluorescent flicker (and solved by setting the shutter speed) or bad white balance - to correctly set the white balance with a good camera shoot raw and adjust in post or crate a custom white balance based on a blank white screen, with an iPhone (or a P&S that doesn't support custom white balance) focus on an area that is pure white or neutral gray (to also control brightness it better be middle gray)

I think that's it, if there are other things that can go wrong or anyone knows how to set the shutter speed on an iPhone leave a comment and I'll update the answer

You can find a more detailed version of this answer on my blog at photographing screens