Carprice. SE - car tires pressure

A little, but not enough to be of significance, and not enough to measure. Think of squeezing a balloon. Adding weight to the car adds to the force squeezing the tires, so pressure will go up a tiny amount, but not as much as the increase in pressure from higher temperatures during the day. If you plan on carrying an unsafe amount of weight, you'll get a higher pressure increase from the heat generated by the flexing of the sidewalls, and from the tread squirming across the road surface.

What's more important is the pressure in the tire determines how much weight it can support. The numbers on the door sticker are supposed to be sufficient for the weight of the car and the maximum legal number of occupants and cargo. Increasing the tire pressure above what's listed on the sticker will increase the load the tire can support, but that doesn't help the car's suspension system. You'll still have degraded handling, steering response, and braking ability. The most pressure a tire should be inflated to is listed in very small numbers on the sidewall. The pressures listed on door stickers are well below those maximum pressures, mostly for comfort, but you also have to consider where you'll buy your next set of tires. What you have now might have a maximum rating of 44 psi, and they'll perform just fine at 40 psi, but 40 psi can't be listed on the door sticker because there are still places where you can buy tires with a maximum pressure of 35 psi. If you inflate those to 40 psi, obviously that's a dangerous situation.

Also remember tire pressures are listed for cold tires. The temperature will be higher, and therefore the pressures will be higher, after a run at highway speed. Do not lower the pressures because of that. The tire manufacturers take that pressure increase into account when they design them.