I got a HD TV last year, because I had the money for one. It's nice (samsung LED Series 5, fairly big) and it certainly makes watching movies and TV shows and documentaries a lot better, but I don't think it improves games all that much. Besides, I hook my PS3/360 to a computer monitor, which has better resolution (although the picture isn't as big) and I advise anyone who cares about resolution to do the same.
But SD suited my gaming purposes for years and years and never caused me any harm. Indeed, keeping an SD TV around can be good - PS1 games look AWFUL on HD TVs, believe me. FFIX looked SO BAD on my HD TV, I had no choice but to repurchase it for the PSP on PSN, where it is at least bearable.
If you don't have the money, don't upgrade. It's not worth being in debt for. And if you do have the money, by all means buy a HD TV. Just make sure it's a good one.