Firstly let's be clear about the difference between a product listing and product info. When you see a number of products listed together, that's a product listing. When you click on one of them, you will be taken to the product info page.
To turn off weight on the product listings pages, login to your Admin area and then choose Configuration > Product Listing > Display Product Weight and set it to 0.
Well, that was easy. You may have spotted the Product Info entry on the Configuration menu and already be there to do the same for product info. STOP.
Product info works differently from product listings. This is because of the flexibility that Zen Cart gives you to define different product types. You may wish, for example, to display a weight for a book, but not for a downloadable pdf. Zen Cart allows you to set them up as different product types and display different information about them.
So where do you go to turn off display of a products weight? Still in your Admin area, you go to the Catalog/Products > Product Types menu. Click once on the relevant product type (usually Product General unless you have taken explicit steps to use another product type) and then click on the "edit layout" button to the right. Then set Show Weight to 0. If you wish to also do this for other product types, go back to the Product Types screen and repeat.