The Battle Master fighter's Lunging Attack maneuver (PHB, p. 74) increases reach of a weapon by 5 feet. Could you choose to increase the short range of the net from 5 feet to 10 feet? This would allow a character to attack with net at 10 feet and avoid close range disadvantage.
Do you forgo adding superiority die to the damage because there is no damage roll with a net, or does the net cause damage when using a combat maneuver?
The description of the Battle Master fighter's Lunging Attack maneuver says ((PHB, p. 74; emphasis mine):
When you make a melee weapon attack on your turn, you can expend one superiority die to increase your reach for that attack by 5 feet. If you hit, you add the superiority die to the attack’s damage roll.
Looking at the Weapons table, the net is listed under martial ranged weapons. Thus, the weapon's normal use is for ranged attacks - which Lunging Attack can't be used on.
If a character wants to make a melee weapon attack with a ranged weapon, it is treated as an improvised weapon. When you attack with an improvised weapon, it's treated as having none of its usual properties, so you'd just be trying to hit someone with the net itself rather than trying to restrain them. (This case is covered in more detail here: Can you make a Melee Weapon Attack with a Net?)
The battle master maneuver affects your reach with a weapon. This is a property separate from the range of a weapon. The net, when used for a ranged attack, uses its own range, but is unaffected by your reach, which only comes into play when you make a melee attack (see PHB 195).
You could, technically, apply the maneuver if you treat the net as an improvised melee weapon, but you would forfeit its special effects that way, as discussed here.
