What PSI should I keep on my 31x10.5x15 tires?

Correct tire pressure is absolutely determined by the year/make/model of vehicle AND tire size and yes, the more air you put in a tire the greater the load it can carry (up to its maximum).

Take your 31x10.50R15 LT 109S tires for example. According to the Rubber Manufacturer Associations Tire Load and Inflation Tables (the inudstry standard) at 30 psi this tire supports a load of 1,595 pounds, at 35 psi that increases to 1,765 pounds, at 40 psi it is 1,945 pounds (compared to a standard S10's P235/75R15 105S sized tires 1,940 pounds at the factory recommended 32 psi) and it maxes out at 2,270 pounds at 50 psi. It is this last number: 2,270 pounds at 50 psi that you will find on the sidewall of the tire, indicating both the tires maximum load and maximum inflation capacity. This is not a RECOMMENDATION, just a safety warning.

So JBF was correct, when you change tire size you need to change inflation pressure to compensate for the change in load capacity. You can see from those figures how overloaded your tires would be if you used the same inflation pressure as the tire size that was put on the truck at the factory.

Source(s):

Data collected from the RMA, Tire Industry Association (TIA) and Bridgestone/Firestone's Tire Resource Manual