After my irises bloom, their leaves begin to fall over. Do I need to trim them back or leave them?

Iris leaves typically droop as they elongate. You should only remove dead or dying leaves (if they are yellowing/browning). The only time you should trim leaves on iris is when you are transplanting them (this keeps the top heavy leaves from falling over while the plants are trying to root-in). If you cut the leaves off, you are removing their ability to continue making food for the next season's blooms.

I have an iris question while we're on the subject. Mine never bloom. The foliage is very healthy and they continued to spread every year, but no blooms! My soil is slightly acidic. What can I do to encourage flowers?

Iris like somewhat alkaline soil.
I have the worst black gumbo clay in the universe, & iris bloom beautifully here. I'm grateful for iris!

Either they aren't getting enough sunlight to bloom (performance for bearded iris will be limited with too much shade), or they've become too crowded over the years and need to be dug, divided, thinned, and replanted. It's also possible you could have problems with iris borers. You may want to dig up some of the plants and check the condition of the rhizomes.