It's hard to say for all vehicles since there is no way to know the weak points. Overloading a Corolla might make the bearings die, and overloading a Kia might snap a control arm.
Of course those are extreme examples (except the corolla... they weren't over-engineered in the bearing department) but there is only so much you can do. There is a buffer since manufacturers know that people will be people. If your GVW is 3000 lbs, the suspension won't crumble at 3001 lbs.
Go to your tire manufacturer's website or TireRack.com and look up your tire and its max weight capacity. Multiply by four. That is what they can technically carry at their max psi. Here is the main caveat. Vehicles designed for higher weights (like pickups) use tires that can perform well over a wide range of pressures. Your passenger tires are designed for a smooth quiet ride, and they are designed to operate at the pressure listed on the door jamb sticker. With a truck, you can pump them up to X psi and they'll do well over a wide range of weights. Passenger tires do not. You'll have to find a pressure that suits your most common weight range to match the weight where the car spends most of its time knowing that it will be too much pressure with no cargo and not enough at full cargo.
Brakes should likely be upgraded. If you're just stretching things a little in the weight category, some good pads and rotors should do the trick.
Spring rate is the correct way to increase capacity. Don't do airbags or air shocks. Shock mounts are designed to just handle the forces that are kicked back from the dampers, not designed to hold weight. Often times they are just bolted to thin stamped steel under the body. Airbags work great until they don't. There is a much higher probability of blowing a bag or developing a leak. The good news is that you can increase spring rate without changing springs. Use some coil spring spacers to remove one round of the coils from active duty. They're cheap and they will eventually fall out, but just replace them.
Shocks should be upgraded as well. Factory shocks are designed to control the oscillating weight of what they are designed to carry. More weight means more up and down inertia.
Automatic transmissions in small cars are not really up to the task of additional weight, but you can add a transmission cooler. Also consider (depending on the car) a computer reflash for higher line pressures and firmer shifts. Manual transmission cars have clutches and diaphragms also designed for their intended weight. Consider an upgrade.
All of that being said, there is nothing you can do about a bunch of other things. Bearings, suspension arms, or the sheet steel they're bolted to. Whatever you buy might be gloriously overkill in those departments, or they could be barely adequate for their engineered GVWR
To do it right takes an engineer and a ton of time and money. The real solution is to just buy something that will take the weight. Doing all of the above will help how the car actually (safely) handles the weight, but at the extremes (panic braking, evasive maneuvers, etc) it can't help if a control arm snaps, a brake caliper pin shears off, or the control arm rips the sheet metal. That is an extreme example, but there is no real way of knowing without knowing what car we're talking about.
If you're talking about a couple oversized people and 10 cases of beer, just buy what you want and drive it. If you're talking about a family of 5 plus an engine block in the trunk, just get something that will take the weight.
Just about a month ago I was being followed by an S10 pickup that had a pallet of bird seed in the back. When he wasn't paying attention and I slowed down for a left turn, he totaled his truck on a bridge guardrail. He couldn't stop, tried to swerve to go around me on the right, the tire pulled off the rim, and he spun into the bridge. All you could smell was burned brake pads and anti-freeze. Fortunately he was fine, and the birds feasted at that intersection for weeks.
Also keep in mind that the GVWR of the car is sometimes a wee bit of BS. They start out as carefully engineered numbers based on industry-standard tests, then the bean counters get involved. That Mazda 6 might be engineered so that it could carry 1500 lbs of cargo by the maths numbers, but keeping it conservative at 1000 means less legal and warranty liability. Someone accountant did the math and realized that dropping the GVWR by 500 lbs would likely mean $50k per year fewer warranty claims on bearings, and if someone wrecks at 1200 lbs of cargo they aren't as liable. I say that because there may be a car out there that is already overbuilt, but its hard to know without an engineering degree.
