I think I may have a problem with my PS3, although it's still under warranty so it's not a huge deal if there is. Let me explain, and see if I can solve it now before shipping it off to Sony.

I am getting no signal from my PS3 to my TV -- when I switch to HDMI source, it says that it has "weak or no signal." And it's very odd, as this came on slowly: it started this morning, and by jiggling the cable and the system I got it to work again. This happened several times today -- but ONLY when I would make the game load its main menu (for example, I would quit playing Ninja Gaiden, and when I'd try and go back to the main menu, the screen would just go black and never recover).

So it seems to have gotten worse all day, with me jiggling everying 4-5 times, until finally now it just stopped. I've tried all the main troubleshooting possibilities -- I switched HDMI ports, and neither port 1 or 2 works. I tried using AV component cables, and still the same "weak or no signal" response. I then tried to take the PS3 to a different television, and still got no response. 

Which tells me it's almost certainly a PS3 problem, not a problem with my TV or with my HDMI cable. Does anyone know what this problem is, or how to fix it? I'd love to do this on my own, if possible.  

Hmm. Firstly, check if you can watch something else from the T.V. such as cable or satellite to make sure it's not the original T.V. and to make sure you got it out of the way, sorry to be so conservative but you didn't mention what else you tried on the original T.V. When you took the PS3 to another T.V. What connection did you use?
But through all of what you have posted, I suspect it is the PS3. Might have to send it in. Well, look on the bright side, I geuss it's better to have a PS3 under warrenty die out then your TV. :/

My cable box is working fine through a coaxial input. I don't currently have anything else to try, but that tells me everything is fine. 

The most telling thing is that the HDMI didn't just fade out at random intervals (which would suggest the TV is the problem), but ONLY when the PS3 was loading menus and such. I used the AV cables to plug the PS3 into the other TV.  This combination worked fine a week ago. 

Still, it's an odd problem. Never really heard of anything like it.

Really strange. Seems like there has been a few PS3 problems popping up as of late. Lets hope thats not part of a bigger trend.

Have you tried reseting the PS3 to it's default settings?

Odd. At the moment I have a problem with my PS3 where it completely freezes after ten minutes of gameplay with weird green splodges coming up on the screen or other weird things, but if on the XMB you can still move it and hear the click as you do, then you have to turn it off at the back. Then I can't turn it back on, because it says it is too hot even when it is actually quite cool. When it comes on for the first time again (after leaving for 20-30mins) it does not output a signal. When I try again after that it works but then will freeze quickly. The most play I can get out of it 10mins, but the amount of play does relate too how long it is left. I have tried different cable out the back but it does not work. I think i am just going to send it in to Sony

hold the power button 5
to 10 second s after you hear the 1st beeps until you hear another one that will reset the system so you can try it on another tv

 Whoa whoa, what's this? Why can't I just turn it off, unplug it, and move it to another TV? Why do I need to do all this resetting stuff to try it on another television? 

Next time you turn your PS3 on, hold down the power button, this should reset the HDMI "handshake" between your PS3 and TV.

 EDIT: ^^^Damn you beat me too it. :(

But if the problem is not with the connection (he tried AV too) then what good will resetting it do?