Short answer: a little, but not as much as you might hope. Rounded edges are a smart design detail, but they are not a magic shield against a car.
The main benefit is that rounded corners are less likely to snag a bumper, tear clothing, or create a sharp puncture point during a low-speed hit. They can also reduce scrapes and make the can slide or deflect slightly instead of catching on the car. That can lower secondary damage to the vehicle and make the scene a bit safer for pedestrians.
But when a car actually hits an urban trash can, the bigger factors are speed, weight, material, wall thickness, mounting method, and location. A fixed steel can may bend or crack and transfer force into the ground or vehicle. A lighter plastic can may fly away and split open. Rounded edges will not stop dents, broken hinges, cracked bases, or crushed bodies.
So I would treat rounded edges as a useful safety and cosmetic feature, not a crash-protection strategy. If a city really wants less damage, it should focus on visibility, reflective bands, sensible placement, impact-resistant materials, replaceable modules, and physical protection such as bollards or raised curbs. Rounded edges help at the margins, but they do not make a trash can car-proof.