Games that you play as a left handed.

The Legend of Zelda series.

you can use a CS:GO command for left handed weapons.

also, non first person games don't really have a left handed option since there would be no difference

F...ire Simulator

In CS:GO, open the console and type
"cl_righthand 0"

3D Sex Villa :3

Forgot Super Mario
Ha destroy blocks with his left hand.

Dark Souls 2 lets you hold weapons in left hand, but I haven't checked if shields work in the right hand.
EDIT: I just fired up Doom, the sprite of hand holding a pistol is definitely left hand.

Talking about Fistful of Frags, it just got updated

Wait, do "normal" left-handed people use a mouse with their left hand? I'm left-handed and have always used it with my right.

EDIT: Nevermind, I misinterpreted the OP and title.

fap simulator 3000, with invisible VR headset.

Die Hard: Nakatomi Plaza

In Clive Barker's Undying, you play as the left-handed protagonist, Patrick Galloway.

Skyrim

Surgeon Simulator.

I think you're ambidextrous in Starbound (?)

I think in Mario Golf 64, you're able to choose whether you want to be right or left handed. If I remember correctly, Mario Tennis 64 had the same feature.

Masterba......laster?