Why should (or shouldn't) we wash rice before cooking?
Reasons to wash your rice:


Reduce/Control Starch levels


Often when you're cooking rice you want distinct grains of rice and for your rice to have texture. In the case of Chinese fried rice for example, you specifically want your rice grains to not stick to one another. If you're talking white rice especially, there will be a lot more loose starch that will form a thickish paste if if you don't rinse it away.

To clean out impurities


I imagine there was a time when there were a lot more impurities (dirt, dust, bugs, etc...). I don't know if that's necessarily the case these days with modern manufacturing. If you're concerned about your source, then this may be a factor for you. There's also the occasional random article that suggest rinsing to reduce levels of something like arsenic (see FDA warning on arsenic in rice)... I think 1. is your bigger factor here though.



Reasons to not wash your rice:


It removes nutrients.


This is true for fortified rice. See here for more information on the fortification process. Note, whole grain/brown rices are less often fortified (I want to say they're not fortified, but I actually don't know for sure). So it would depend on what kind of product you're buying and where it came from.

You actually want to keep the extra starch.


This is the case for things like risotto where the starch is what gives the dish it's creaminess. Serious Eats has a great article that talks about the process.