Do airlines raise the ticket price as flights fill up?
I know YMMV, but as a general rule, is it how they operate?
and as a bonus question, once a flight is full, would they schedule a new one on the same date with lower prices at first?
Airlines publish different prices in different fare buckets. As soon as the cheaper fare buckets are sold out, the more expensive ones are automatically used. So to your first question, yes, the flights get more expensive when they fill up.
To your second question: Flight plans are usually planned for half a year, summer and winter. If a flight is full, then it's full. Usually an airline doesn't have the capacity to schedule another flight on short notice. They would have to hold replacement aircrafts and cockpit crews just for this rare case, which costs a lot of money.
There are two things in effect:
Fare classes, as explained by dunni's answer - the tickets are divided to classes at different prices, and the cheap ones are sold first. As the plane fills up, the cheap classes are sold out thus prices go up. Note that fare classes don't imply different service - people buying cheap and expensive tickets will all get the same seats, food etc.
Fare changes based on demand - during the months before a flight, the airline can change the prices. If ticket sales are below expectations, the airline may reduce prices in attempt to sell more tickets. If they're above expectations, they may raise prices to maximize profits.
As a result of these, when a flight's date is near, two different things may happen. If ticket sales are normal or above, only expensive tickets will be left. If sales were slow, the airline may offer cheap last minute deals, trying to fill the plane.
Each airline has a yield management department, whose whole job is to adjust prices to maximize revenue or profit as information comes in. They consider events in the area-if the Olympics are scheduled in an area prices near there will be raised. They consider demand-if tickets are selling well for some place prices will rise. Who knows what else they consider? Prices rise and fall based on their assessment. They will aso move seats between fare categories. I have seen claims that on average you should book flights a certain amount of time ahead, with long haul flights further ahead than locals. Whether the people asserting that know what they are talking about is for you to assess. Maybe they have done a careful statistical analysis of airfares over time. Maybe they are making it up based on personal experience. Maybe they are repeating what they have heard or read somewhere. All of this is based on stuff I read somewhere, so you can assess its validity as well.