I have a one step process...
Step 1: pour directly into tubes with no spilling.
Just kidding. I scrape mine with a plastic thingy, use a paper towel to wipe out any holes. If I see any spots that are particularly difficult, I will hit it with my hot air gun with the tray placed on paper towel. Wash in hot water with Dawn.
If they still feel sticky or slick after that, I spray with iso alcohol and wipe with a paper towel.
Still sticky? I throw it in my dishwasher. (really, any oil/wax that's left is going to be minimal by this point. There's more oil on my cooking pans than on this lip balm tray at this point.)
I have seen some videos where people do the scrapey thing with the filled tubes still in the tray. I guess it kind of makes sense then that the overspill wouldn't get pushed into the empty tray holes... I just can't bring myself to do it. The naturally rounded tops on lippies is too satisfying.
Occasionally they're a pita to clean, but I wouldn't make lippies without one.