There are a bunch of trees that have red leaves either from day one, or turn partway through the summer: * Schubert choke cherry. Leaves are green for the first half of the summer then go red. * Rosybloom crabapples. Have leaves ranging from greenbronze to purple. * Diablo, Centerglo, and Amber Jubilee ninebark. My working theory is that there is a gene that creates red/purple pigments, and that the homeobox that controls the expression of that gene is faulty, so the tree keeps producing it. I would think that this gene would be hard on the plant. The excess red would interfere with photosynthesis, and the plant would be less competitive than it's green siblings. A quick internet search leaves me unenlightened as to the details of the cause.
Red plant pigments (anthocyanins) may confer some advantages, too; some that have been suggested include ultraviolet light protection, thermoregulation (absorbing solar heat in cold air), and reduced herbivory (herbivores don't see red / brown leaves as optimum food, mistaking it for sick or dead leaves).
Ok. Granted, although I'd not thought of either before. I don't know the UV response of the anthocyanins. Most grazers don't see colour. A red leaf may be darker. Deer at any rate, aren't very visual. (And frankly aren't too bright. How many smarts does it take to stalk a leaf...) However there are very few species that are consistently red, so overall I suspect that it is a disadvantage, until you convince some primate that it's worth fostering for novelty value.