How can I see the GPU temperture of my ATI graphics card?

If you are using the proprietary driver (fglrx), then the command you want is aticonfig --odgt.

So far as I'm aware, for many cards this sensor will not be exposed through lm-sensors.

Not long until it fries. Depending on the amount of graphic workload you're doing, manufacturing process, accumulated dust could be minutes, hours. buy an aftermarket fanless cooling replacement if noise is a problem.

To monitor your GPU Temperature, use this Q&A to help find the GPU sensor:

How to use lm-sensors?

Firstly, (See http://www.unixmen.com/howto-install-ati-display-driver-in-ubuntu/ to install) then type

sudo aticonfig --initial

Then

aticonfig --odgt

OUT:

Default Adapter - ATI Radeon HD 5700 Series
                  Sensor 0: Temperature - 45.00 C

sudo apt-get install lm-sensors
sensors

Which outputs something like:

radeon-pci-0100
Adapter: PCI adapter
temp1:   +48.5°C  (crit = +120.0°C, hyst = +90.0°C)