When to lower the axle on a tri-axle dump?

When you're loaded to the point where your weight will exceed your legal weight as a tandem truck. Honestly, you should probably have it lowered any time you're loaded, since a motor carrier officer who sees a tri axle, quad axle, etc. with a raised lift axle is likely to question it... even if you are legal, it's time wasted and another annoyance to deal with.

If you mean when after you get loaded, you do it while the vehicle is not in motion - don't lower your lift axle while the vehicle is moving.