Do multiple healing effects stack?

Lucio's aura

Does not stack. You can have both speed boost and healing songs at the same time though, and Amp it Up can be staggered by different Lucio to provided the stronger heal for longer period of times.

Mercy's beam

Yep, stacks.

Soldier: 76's zone

Does not stack. Again, it's better to stagger them to provide a longer lasting Biotic Field.

Zenyatta's orb

Does not stack.

Any two different kinds of healing stack, and basically only Mercy's Caduceus staff stacks with itself, healing-wise.

I tested the Lucio, Mercy, and Soldier: 76 interactions myself during normal play. Also helps that my own observations and the Wiki's do not conflict with each other.