Most likely, reapplying the thermal paste will not help. The issue is not that the PS3 is overheating - at least, not any more. The issue most bulky PS3's are having that causes YLOD is that the solder connecting the BGA CPU/GPU sockets to the motherboard becomes cracked or warped. Many XBox 360's are having the same problem - the issue is the lower melting-point of the lead-free solder used in modern electronics (leaded solder is now illegal in some European countries).
How can you tell if this is your issue? If your PS3 can play games for a while before shutting off, it's likely overheating, and as @aslum said, reapplying the thermal paste should help. However, if your PS3 seems to start for a second, but then shuts down, and never displays anything on the screen, this is the dreaded YLOD, caused most likely by the issue I mentioned above.
There is still hope to bring your PS3 back, though, at least long enough to get your data off. You need to do exactly what the circuit-board manufacturers do: remelt the solder-joints to form solder-balls, then re-cool them to reform the connections - this is called reflowing. Ideally, you'd like to use a professional reflow-oven; however, these cost tens-of-thousands of dollars, so unless you know someone who works for a circuit board manufacturer and can get you some time in their reflow oven, you'll either have to send it in to a professional, or do it the less reliable way: in your home oven. (though this is still much more reliable than the method others recommend: using a heat-gun. That will be much faster, but do a much poorer job and probably cost more)
The above link gives detailed instructions on reflowing a PS3 board in a home-oven. However, before you do that, some caveats:
- Follow the directions very carefully! I cannot stress this enough!
- Reflowing is a very finicky process, and if you heat your board even a few degrees too hot or cold, or cool it down too fast or too slow, it won't work and could even wreck your board. Also, if you don't properly protect some of the components, they will melt.
- Many home ovens do not display accurate temperatures, so getting a separate thermometer (as recommended in the article) is a must.
- Even small amounts of moisture on the board can cause it to warp and wreck the board when reflowing, so do not skip the pre-bake step. This step takes 4-8 hours, so allot yourself some time. But remember, you should never keep your oven on over night while you are sleeping (for any reason).
- Make sure your kitchen is well-ventilated, and you have a self-cleaning oven. Both solder-flux and the putty mentioned in that article release toxic-fumes. It's nothing that's going to kill you, but it's also not something you want lingering in your kitchen, or especially your oven. Keep your kitchen well-ventilated for the whole process (and probably several hours afterwards), and flash-clean your oven immediately after you are done.
You'll need about $10-20 worth of tools, your home oven, and a full day's worth of time (most of which is spent in the pre-bake step; again, don't skip this step!). Good luck!!