In zencart is it possible to turn off shopping cart line that says "Weight: 0lbs" in "Total Items: 1 Weight: 0lbs Amount: $38.95"?

Yes.

Go to:   Admin->Configuration->Shipping/Packaging
Select  Display Number of Boxes and Weight Status
Choose the desired setting:
0= off
1= Boxes Only
2= Weight Only
3= Both Boxes and Weight