Yes, maybe, but only rarely…
General consensus is that because a Net deals "–" damage (no damage) rather than 0 damage, other features can't add additional or extra damage to it. Designer Jeremy Crawford summarises this as:
"The intent is that a net doesn't deal extra damage because it's not dealing damage in the first place."
So in order to add damage (from other class abilities/feats) to Net attacks, you would first need to swap out the "–" damage.
Here are ways to potentially make a Net do damage…
Cast Lightning Arrow
The bonus action Ranger spell Lightning Arrow (available to Rangers at level 9, and also to Lore Bards at level 6 through Magical Secrets) significantly uses a different wording to most similar spells:
"Make the attack roll as normal. The target takes 4d8 lightning damage on a hit, or half as much damage on a miss, instead of the weapon's normal damage."
Both multiclassing with a Rogue dip to add Sneak Attack damage, and taking the Sharpshooter feat to add the +10, should work on this once the Net is inflicting damage, and all it took was 6 levels of Bard.
Use Monk Deflect Missiles
Have a friend make a ranged weapon attack with a Net at your level 3 Monk and use Deflect Missiles to try to catch it, and then 1 ki point to make a ranged attack with the same Net "as part of the same reaction".
"You make this attack with proficiency, regardless of your weapon proficiencies, and the missile counts as a monk weapon for the attack."
Your Monk can now inflict damage with this Net attack because their Martial Arts feature says monk weapons deal d4 damage "in place of the normal damage" rather than "extra damage" or whatever.
This Monk approach depends on your DM ruling that you can "reduce the damage to 0" with Deflect Missiles for "the damage you take from the attack " despite a "–" damage dealt by the Net (which may be related to the "Taking 0 damage is the same as taking no damage" point here https://rpg.stackexchange.com/a/111864/81166); and that a Net is "small enough for you to hold in one hand", which could be argued as a Net is thrown with one hand (and Monks are meant to be dexterous). For the counter argument, see: https://rpg.stackexchange.com/a/145088/81166.
Use a Ranged Weapon to make a Melee Attack
Finally, you could do damage with a Net by giving up (some or) all of the Net's features.
The rules in the Improvised Weapons section of the Player's Handbook say:
"If a character uses a ranged weapon to make a melee attack … it also deals 1d4 damage."
So by using your net to make an improvised weapon strength-based melee attack it will do damage, but almost certainly without retaining its special properties to restrain on a hit.
Narratively this could make sense as some nets were weighted.
The Tavern Brawler feat would make you proficient with these improvised weapon melee net attacks.
It's very debatable whether the net is still a 'ranged weapon' for the purposes of Sneak Attack or Sharpshooter (although "a character uses a ranged weapon to make a melee attack" could imply that), but you could add other class abilities like Rage bonus or Battle Master Maneuvers for example. If you still want to restrain with an improvised melee Net attack, then cast Ensnaring Strike (1st level Ranger spell).