In another forum someone said "Screen down the fillneck may be partialy plugged up with matter." Is this something a neophyte can change or fix himself, or should I take it to an auto shop? How can I get to this screen to see if it is clogged or blocked?
This is probably something you should take the car to a shop for. It sounds like a problem with the evaporative emissions system.
Your tank is a completely sealed system and when you are pumping gas it needs a way to vent the gas fumes from then tank because the gas flowing in pushes the fumes out. There are vent lines and solenoids that run to a charcoal canister which is usually in the front of the car. If there is something wrong with one of these solenoids or the charcoal canister itself it, will prevent the fumes from escaping and could cause the problem you are describing.
I finally took it in to a Toyota dealership. They said it was the overflow control valve (OCV); it's inside the gas tank, and requires replacing the entire gas tank, about $1000.