Example: Using Grass Knot on a pokemon with a Lagging Tail or something.
There are no items that increase the weight of the Pokemon. [Source]
The only item that effects weight is the Float-stone and that decreases weight.
There are two abilities that effect weight Heavy metal and Light metal which double and halve weight respectively.
While the Lagging tail's description does say that it is tremendously heavy it does NOT increase weight.
The Iron-ball lowers speed by 50% and grounds Flying types but does NOT increase weight.
So to answer NO Items do not effect weight related moves like Grass knot and Low kick, however Fling is effected by the item and one can assume that Iron-ball does the most damage because it is Heavy.