How do I reduce the risk of pinch flat punctures while riding up stairs on my hardtail? I've got a Specialized Pitch, size 27.5". I do a lot of urban MTBing and tricking.

I know how to ride up stairs - up to four stairs at the moment, working on getting more - you lean back, get the front wheel up the first three or so, then lean forwards and pedal, keeping the weight off the back wheel. I work up a bit of speed and then pedal up, with gears set at 1 in the front and 3 in the back (I've got 3 in front and 8 in the back). The back wheel is at ~40 PSI.

But consistently, I'll make it up this staircase ~5 times in a row, and then, at some point, inevitably, I'll get a pinch-flat puncture.

This has been the main expense on the bike recently - this one staircase has cost me hundreds in replacing the inner tube.

Is there anything I can do to avoid getting these punctures? Am I simply not doing it correctly?
I used to get snakebites like crazy doing urban riding, even running really high pressures. I'm a big guy. What finally helped was running higher volume tires and wider. See if you can fit a wider tire in your frame, and look at getting some wider rims. 

Increasing the volume of air adds more cushioning. As Batman mentioned, going tubeless also helps, since the tube displaces some of the volume between the rim and the tire.

If you're sticking with tubes, get some DH tubes. Normal tubes are around 1mm or less, DH are often around 1.5mm. They will weigh more, but might hold up better.