Have you tried different cables? Instead of HDMI cable, try to use component cables or the other way around. If the XBox is still under warranty, call and they'll send you out a new one.
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This could be a number of things, but definitely start with alternate settings on both your tv and 360, trying different resolutions, etc. 

Hopefully this will work for you, but this symptom has also been showing up often as a prelude to the E74 error.  This is most likely a failure of the 360's scaler chip, but can also be caused by gpu failure.

Lastly, check your power supply 'brick'.  Try to minimize interference, (wires/cables running close together) and make sure the power supply is ventilated, and not laying flat on carpet.  Try using different power outlets, if using a power strip, try plugging directly into the wall socket.

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Check this thread:

http://www.xbox360forum.com/forum/digital-home/60195-red-lines-screen.html

According to one poster, it may have to do with improperly grounded wiring. 

I hope this info helps! :)
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