No, it doesn't generally increase the weapon's damage
When you cast Heat Metal on an object you get the following effect, emphasis mine:
Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell. Until the spell ends, you can use a bonus action on each of your subsequent turns to cause this damage again.
Anyone touching the object only actually takes damage from it once immediately when you cast the spell and thereafter when you use a bonus action to cause the damage again. Otherwise, touching the object (despite the heat) does not cause damage, so simply hitting someone with such a metal object wouldn't do any more damage than normal - you don't have to worry about accidentally giving your foes a boost by using the spell against them.
The above said, there are a couple of ways I would allow you to get this kind of effect if you were deliberately trying to.
Firstly, if you readied the spell to be cast at the moment the target object was used to hit someone, I'd rule that both the wielder and their target were both in contact with the object at the time of casting, so both would take the damage.
Secondly, if you were to cast it on a weapon you were holding yourself, then on subsequent turns you could in principle use your bonus action to cause the extra damage exactly when you hit someone with it (since a bonus action without timing specified can be used whenever you like on your turn). That would get you one hit per round with some extra damage.
Obviously not a recommended strategy for use on yourself/friends unless you've got fire resistance or immunity.