Any idea what this part is? Fell off my wife's 2003 Neon SXT.

 

Looks like a heat shield,cat, exhaust system.Get it in the air(lift) and take a look.

 

So, I believe it is the shifter cable heat shield based on some searching. How important is this?

 

Here is my theory about things like this. If a manufacturer could save even a few pennies on a car by not putting a part on a car, they would have not put it on the car. That being said, I don't think you will notice any adverse effects regardless..