[removelineitem cart=cartID&index=3]

Putting [removelineitem] in your template immediately deletes the specified line item from the specified shopping cart file. See also [addlineitem]. If the cart file is not in a ShoppingCarts folder, you may use the alternate form file=/folder/cartID instead of cart=cartID.

Take caution when removing multiple items. As you remove items, the index numbering changes. When you remove item number 2, the subsequent items move up one. For this reason, you need to figure out a way to adjust the index number. Decrement the index each time something's deleted, such as the following:

(In this example, in the incoming form the cart items each have a delete checkbox named kill_[index]. Note that in a math context, you do not need the square brackets within the expression.)

[math show=f]adjustby=0[/math]

[formvariables name=kill_&exact=F]
[math show=f]deleteitemnumber=[getchars 
start=6][name][/getchars]+adjustby[/math]

[removelineitem file=ShoppingCarts/[cart]&index=[deleteitemnumber]]
[math show=f]adjustby=adjustby-1[/math]

[/formvariables]