I have a similar car to the OP, a 2007 Corolla with the 195/65R15 tires. I bought it used with 16k miles and it had the OEM Bridgestone Insignia SE200 tires on it. They were all fairly worn, and I picked up a sidewall puncture in one near the 19k mile mark, so I just replaced them all. I can't comment on fuel economy with the Bridgestones, as I was still feeling out how the car wanted to be driven, and I was also having fun with it at the same time. I replaced the Bridgestones with Yokohama Avid TRZ tires.
I've tried a lot of different pressures with these tires on this car. The Yokohama tires tend to really harden up when cold, so I'm always playing with tire pressures in the winter. Sometimes I'm into the crisp ride, sometimes not. I track fuel economy on every single tank and input it into
www.trackyourgasmileage.com, so I know every tank that I've put in the car. I must say that I haven't really measured any significant difference in fuel economy with the different tire pressures. I've ran as high as 38 PSI in the tires, and as low as the placard-recommended 30 PSI. Right now, I've settled on 32 front/30 rear. I almost always get right about 35 PSI with this car. My overall average for it is 35.35 MPG.
I've read some conjecture on the internet that as you increase pressures high enough, you start to increase your rolling resistance because the dampers on the vehicle will have to move rather than the tire absorbing the texture of the road. Conceptually, it makes sense. Imagine a 50-pound solid rubber wheel that you try to rotate over a brick; this would represent an excessively-inflated tire. You have to lift the entire 50-pound piece of rubber to cross the brick. Then imagine a 50-pound rubber tire/wheel combination inflated to, say, 20 PSI, representing a normal or even under-inflated tire. It'll still take effort to move it across the brick, but the tire will deflect allowing you to push the weight forward rather than lift the weight up over.
Assuming this concept is applicable to a vehicle on an actual road, there is necessarily going to be a gray area as you raise air pressure to lower resistance. Eventually, you'll reach that point of diminishing return, and may actually cross the line and begin to increase your resistance as you go too high. I imagine that with the VAST array of different vehicle weights, tire sizes and construction methods, road types, etc, that line (or even gray area) will be vastly different for everyone. Like I said, I've personally not found any significant delta in fuel economy that I could contribute solely to a tire pressure change. Objective testing shows a nominal effect, and a diminishing effect as you increase pressure much over 35 PSI. I fully expect everyone to have different results here, though. Somebody might gain a significant amount by going from 35 to 40, while others don't even see a difference between 30 and 40, let alone 35 and 40.