Well.. now we get to the TimeCurve part.
If you go to the  button and you click (or slide) on the left of it, it will change to the IPO editor.
Here you can change allmost everything in time with a curve.
Oke.. now the button is activated.. well thats for animation/motion curves.. well,
thats not what we want to change..
We want to change the Material settings in time..

So we need to go to the   button to get the Material IPO .

Well.. on the left and bottom of the window we see allot of numbers
The numbers below are framenumbers.. (that the timebar if fact :)
The numbers on the left side of the window are the values. (just like SIZEY: 0.010 <- those values ;))
On the right side of the screen we see the Material settings (it allmost must look familiar to you by now,
because they are the settings of the Material Editor =).
Click once on `Alpha '.
So we select the function of the Material settings that we wanna change.
 

With CTRL-Left mouse button you can set points for the curve.
Add one at Frame 0, Value 4
Now add a point at Frame 100 , Value -4

This should fadeout your fire nicely (i hope), because i haven't figured IPO Editor out yet that well ;).

Well.. i'll say.. play with the settings to find out what they do,
How we say in Dutch 'You'll learn while playing' =) (oke... maybe it is very Wrong translated.. but hey.. i aint an enlish dictionary ;))..