How do I prevent honey or brown sugar from burning to the bottom of the pan when I'm using them as a coating on roast meat or veggies?

You might also be able to mitigate the problem somewhat by using a glass baking dish or, if you're doing large quantities, a sheet tray nestled inside a second sheet tray, and setting your oven rack to a higher position.

When sugar syrup is exposed to high temperatures, first the water boils off, the sugar caramelizes, then it turns to pure carbon.

Sugar -- Water + (Time * Temperature) = Over Caramelization aka Carbonization aka "Black"

So… Dilute the sugar with water, reduce the time or reduce the temperature.

The more you dilute a sugar solution, the longer it will take for the water to boil off and for caramelization to begin. So that's one method. And so is Sarah's suggestion. But I happen to know (because I am quite fond of the technique) that you can successfully roast squash in undiluted maple syrup without burning it. So I'm guessing either you've got your oven cranked up too high or you're cutting your vegetables into pieces too large to cook before the syrup gets too dark.

If it's affecting the taste of the dish because it's burning or over-caramleizing, add it halfway (or 2/3rds of the way) through the cooking.

If it's because you don't want to clean the pan: place the goodies on tin foil and discard the foil after baking.