Why am I seeing red dots on my LCD screen?

My laptop is about 2.5 years old. Now I am starting to see red dots on certain shades of colour (mainly dark colours, blues and blacks), and it is not limited to certain pixels, because when you move a window around, the red dots move with it, staying on the certain shades of colour.

Is this a problem with the LCD screen, or is it the GPU? Is there a way to determine this?

It is clearly not a driver issue, because it happens in Linux and Windows, and my Windows setup has not changed prior to the issue starting.

Sadly, this sounds like bump underfill cracking on the GPU. It will get worse and worse and eventually it will destroy the machine. There is one way to prove it conclusively -- does the problem tend to get worse when the laptop is cold and get less serious as the laptop warms up? (Or did it start out that way?) If so, it's definitely bump underfill cracking.

As the GPU heats up and cools, the components that join the GPU die to the motherboard expand and contract at different rates, creating stresses on the connections between the GPU and the motherboard. Eventually, the material starts to crack, leading to intermittent connections. At least at first, the problem tends to be worse when the laptop is cool because as it heats up, the expansion tends to force the bumps into better contact with the underfill.

It's very unlikely to be an issue in the display because the red dots move with the windows.

For me it was only a bad connection between the monitor cable and graphics card.

I've just had an identical problem. It has been solved by fixing (adjusting/reconnecting) the monitor cable to the graphics card, as suggested above.