PS3 flickering while playing games

You can try turning off HDMI deep colour: Settings -> Display Settings -> Deep Color Output (HDMI). That can cause compatibility problems, especially at higher resolutions.

If that doesn't work try using the AV cable that came with your PlayStation 3. Since it's just a composite cable you'll be stuck at SD resolutions, but give it a try to see if it stops the flickering. Switching to an AV cable after using an HDMI cable is a bit of pain, but you can follow the instructions on Sony's support site. If you go with the first option, holding down the power button, disconnect your HDMI cable first.

If that works, then check to see if your TV has a component video input. If so buy a component AV cable and use that to connect your PS3 to your TV. The official Sony branded cable is fairly expensive, but it's not hard to find cheaper clone cables. That will allow you to use HD resolutions and if the composite cable does't flicker the component cable probably won't either.

You might not be able to get component cables working at 1080p. Many TVs don't support this resolution with their component inputs. You may have to use 1080i instead.

NOTE: This answer would void any warranty

A very common problem with PS3 consoles (especially after years of use) is that they start overheating. This can cause multiple problems, among them are video issues due to failing solder joints near the GPU.

The reason this happens is over a course of time, the solder holding critical components on the Playstation (in your case the GPU), becomes cracked and damaged from regular use. This problem is also more prominent in Playstations that were not given enough room to vent while running, but can happen to all of them.

Since I had the same issue (and used to do professional hardware repair of Video Game consoles) I am fairly confident that a 're-flow of solder' would fix your issue. But it is VERY involved and not for everyone. (i.e. you may just want to buy a PS4 or another PS3 instead)

In order to re-flow the solder you would need the following:

1. Industrial Heat Gun (to heat solder to melting point)
2. Lead-free Flux (to 're-flow' the solder and fix damaged solder joints)
3. Various tools to take apart the Playstation 3 (most of which are security torx bits)
   

NOTE: It is also recommended to have an IR temperature gun to ensure the processors themselves are not getting too hot while re-flowing solder.

As a quick idea of what you would need to do:

1. Take the Playstation 3 COMPLETELY appart
2. Apply flux near solder joints at the GPU, CPU and any other places where you suspect there may be damage (Doing a majority of the system board is recommended)
3. Use the heat gun to heat the solder points until they are 'shiny' and 'melting'. If you got the IR temperature gun too, be sure to monitor the processors according to the Playstation 3 tech specs (which if I find again, I will update this post to include them)
4. Wait for solder to solidify again (wont take but a few seconds after removing heat) and put your Playstation back together.

As I said, I have done the above process on many Playstations with VERY similar issues. I hope this can either help you, or at least provide some guidance on what might need to be done if all other options have failed to fix your issues.

As another note, if you do plan on getting another PS3 and don't want to lose your games (and if your Playstation is still working enough to do it) be sure to back up all your games in a manner you can restore them to the new Playstation. Since you can't just take the existing Hard Drive out of the old one and put it into the new one without issues.

Looks like what happened to a nVidia GPU i overclocked way back in 2005. I'm afraid this could mean you have thermal damage to the graphics chipset in the PS3.

A reflow could solve this if the heat have cracked the solder joints, but not damaged the chips. (Look at @Circuit Static's answer)

Where did you have the PS3 running? In a confined space, or on a couch?

Gaming appliances should always have access to an abundance of air.

Overheating could be caused either from running in a confined space, or by dust blocking the air flowing into or out of the console.

Try using another HDMI cable. had the same problem a few years ago and it worked for me, oh and overheating can also cause this.