Response time doesn't have the effect that some people in this thread are implying. Response time is not input lag; it only describes the amount of time it takes for a pixel to rotate to a new colour value after being commanded by the television to do so. It does not account for things like image processing, which usually takes far longer than the response time.
This isn't to say that response time isn't an important factor in deciding what sort of screen you want to play on; the more time the screen spends rotating colors, the blurrier the motion will look, and playing fps games with blurry motion sucks.
But it is only a single small part of input lag.
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The main source of input lag from a TV is usually the image processing; modern LCDs often have response times of only a few milliseconds. But with LCD displays you're still probably looking at a good 30ms+ of total input lag because the TV has to receive the entire image, and process it to fit and look nice on its fixed pixel grid, before actually sending it out to be displayed.
SD CRTs, by contrast, aren't designed to do any such thing. As chunks of video signal enter those things, they get split up and manipulated by a few filters, and then usually immediately shot out the electron guns. When a line of video starts entering an SD CRT, you can expect the whole line to be burning phosphors by the time a millisecond is up. Hence, they usually have extremely low input lag.
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Does this mean you should necessarily use an SD CRT? Maybe, but not necessarily. 20-30ms input lag is actually quite low, and for most purposes a really responsive, blazing-fast modern HDTV might as more or less as snappy as an SD CRT. Furthermore, SD CRTs display video interlaced; in some ways it's not as damaging to visibility as LCD blur, but it can definitely make the image look somewhat icky when there's medium-speed motion on screen. Another downside is that, if your SD CRT is a 4:3 model, you're probably going to be screwed in terms of Field of View in modern games. Halo 3 is a pretty bad offender; 16:9 users get 70 degrees of FoV, 4:3 users get only 55.
[Edited on 05.18.2012 3:41 PM PDT]