One of my PCs managed to get a +2 Longsword from a kobold tinkerer that experiments in magic for only 200 gp (it was on a remote island and she would have no idea of prices on the mainland)! Since she only experiments in magic, it would make sense that the longsword is cursed in some way accidentally. He is only lvl 2, so having a +2 longsword is pretty OP so I want a good curse that is unknown to him, but is still annoying. Got any ideas? My idea was that every time he crits using the sword, the enemy gets advantage on it's next attack. Can you give me any other ideas?

Ooh, I love myself a curse for a weapon. Perhaps she begins to feel very attached to the longsword, so much so that she has to later begin rolling wisdom saving throws to not cut herself with it (this is of course, an obvious copy to Lord of the Rings but you can implement similar magical curses from other stories as you wish lol). 
Perhaps it begins floating when she sheathes its blade sometimes and flutters aimlessly away and the PCs have to try to catch it? 
Perhaps it lets out a loud shriek that can be heard from far away as it strikes an opponent, used, or even just held.
The opportunities are endless, good luck and tell me what you manage to figure out! I'd like to know what mishap you inflicted on this unfortunate kobold :)

I haven't inflicted any mishaps on the kobold yet, but all i'm saying is, one of her 'experiments' may blow up in her face ;)

Just make it a slight variation on the Berserker Axe, and maybe require attunement so it’s really locked into their soul.

I've a few ideas that are annoying.

You could have the sword draw monsters towards it at night, so every time they long rest, there's an attack, making it more difficult to safely long rest, so even when they get Leomund's Tiny Hut, they'd see and hear monsters circling all night.  This could even cause roleplay conflicts as the guards in cities eventually realize that monsters try to get into the city at night whenever this party is there.

Maybe to maintain that +2 magical modifier, the character loses 1d6 hit dice at the end of every Long Rest (not hit point lost, but hit dice lost, so they're not available when resting for natural healing).

Maybe it begins talking, constantly giving her bad advice in combat?

Or it could have a pre-recorded message, advertising the tinkerer's business... really bad if it goes off when the party is sneaking.

The sword is scretly a Sword of <cretaure type> Slaying... and forces it's wielder to fight <creature type> manically when they are encountered. You don't even have to decide which <creature type> until it's convenient. It's a little ace of chaos up your sleeve.

These are all great ideas!! Thank you all so much. I'll take these all into account

Thanks!

I'd just give it charges. Starts with say 20 charges, and every hit expends 1. They don't regain charges ever. Once it depletes, it becomes a normal sword. 

Or if you want to milk it a bit, once it drops to zero you can have it becomes a +1 longsword with an additional 10 charges. Then rinse and repeat until it's a normal sword again at zero.