@ my Shul, the members of the Chevra Kadisha do their task gratis; It is a privilege to be involved with such a great Mitzwah, and they wouldn’t wish to detract from the Kovod hoAcharon by being paid in this world.
Kohanim can only have token share in this Mitzwah, by cutting planks of wood for a coffin, before the coffin is brought to the deceased, or by helping with Shiva Minyanim.