How might a coral be affected if all its symbiotic algae died?

The coral would either adapt or die. A symbiotic relationship is a relationship in which both organisms benefit. Thus since the coral no longer obtains the benefits of it's relationship with the algae it will either find a new relationship, evolve a new adaptation, or die.

This symbiotic relationship is an example of a mutualistic relationship. Symbiotic algae live within the tissues of a coral. Algae through the process of photosynthesis provide oxygen, glucose and other nutrients for the coral. The coral uses these nutrients for a synthesis of carbohydrates, proteins, and other molecules. On the other hand, algae are safe from the predators and also get nutrients from the coral tissues for photosynthesis.

Thus, if all symbiotic algae die, the coral will probably die. However, it can also find new organisms for the relationship and adapt, but it takes some time.