Tiny insects are crawling out of the drains of our sinks and shower. They mature and become flying insects. How do I kill them without damaging the pipes?

Sounds like you have some drainflies. Drainflies (more properly called phorid flies) lay their eggs in the film that accumulates on the sides of pipes.

The old fashioned folk remedy is to pour a cup of chlorine bleach for every 2 inches of pipe diameter into the drain. That may still kill off the adults.

Most hardware stores these days stock an enzyme drain cleanser specifically designed to dissolve the film the eggs are laid in. Follow label directions carefully to avoid damage to the pipes.