Should you cut down old iris blooms when done blooming?

Yes, the blooms should be cut off iris plants.

Specifically, the plant in question has to redirect energy and reprioritize life-sustaining activities into seed production and subsequent fall and winter survival patterns. A bloom which is not removed provides competition for the above-mentioned concerns. It also will attract unwelcome pests to the plant when it is vulnerably trying to ensure perpetuation of the species.