Does improving weapons make them heavier or lighter?
The upgrades reduce the weight. See below the comparison between the base level Raider I shotgun versus the upgraded Raider V.
Upgrades that "improve" weight reduce weight, such as the SMG Ultralight Materials. Since it's an upgrade it reduces the weight. To my knowledge all upgrades are purely positive.
Lighter. NEVER heavier. Upgrades will always decrease weight as well as increasing damage, accuracy, capacity, and firing rate(for most weapons.)