Dwarf Burning Bushes
Is there some nutrient that should be used in soil of a burning bush to make it a deep red?
Micro-nutrients and trace minerals play a big part in enhancement of color of foliage. I wouldn't think in terms of a specific nutrient, but I believe using a mineral balanced, complete organic fertilizer that includes the minor and trace minerals will be the best thing you can do to improve plant health and natural color, along with compost mulching.
I use Agricolas 4-8-4 and am always amazed at the luster of foliage color, flowering and fruiting and taste of produce. But I haven't grown 'burning bush'.
Next best may be Dr Earth or Down To Earth brands All Purpose.