Can you tell if artwork was made with the Live Trace feature?

You can tell it when the artwork is rather geometric or when has text elements.

I agree with @Doug A Roberts you can often tell by observation. 

I am a teacher and I often have to see if the student used Live trace.

This is what I do:

Analyze if the artwork is a copy of another drawing or illustration. through a google image search. If you found something similar, download it. Place it and apply Live trace. If the result is too similar, then you can tell by comparison.

Usually when you create someone creates from scratch, never use too many paths because it is harder to edit the anchor points.

Not programmatically, if that is what you mean. There's no fundamental difference between the paths live trace creates and paths you could make yourself. You can often tell by observation though.