When should you tint primer?
The answer depends on the type of paint you will be using and the new color. If you are using a deep color, then I would recommend that you use the new Benjamin Moore Aura paint (in either the matte, eggshell or satin finish). This paint covers so well, that you only need two coats total - and no primer. In fact, the Aura paint is self-priming, so you don't even need primer even if you are going over new sheetrock or spackle.
If you aren't going to use Aura and you are using a medium to deep color, then you definitely should use a tinted primer - not white. The tinted primer will help in reducing the number of coats, but likely you will need a total of between 3 and 5 coats including the primer.
If you are going to use a light color, then tinting the primer isn't necessary.
Overall, Aura is not only the least expensive option, it is also the best and the easiest. Colors in Aura will hold up much better than non-Aura paint. The Aura is much more washable and won't burnish. And given that you will be done in about 1/2 the number of coats, you will spend a lot less time painting, too.