There is an orange blooming epimedium-WARLEYENSE.
They like dry shade and the plant is attractive after the tiny flowers have gone by in the Spring. Check out plant delights site for some of them, also Garden Vision. I have ordered from Garden Vision in the past.
Von

You will likely get the best selection of flowers in this color from annuals or tender perennials sold as annuals - Fuchsia 'Gardenmeister' and its clones, impatiens, mimulus (monkey flower) all offer pretty intensely orange flowers and are shade tolerant. Orange or rusty/coppery colored coleus would work also.

Since flowers come and go, you might want to try some of the new heucheras with orange foliage (marmalade, peach flambe) -- perhaps in containers if you don't plan to water the area beneath the trees. Annuals will need to be watered too if planted there. My Princess Irene tulips (orange) have done amazingly well under deciduous trees (returning for the 3rd year this spring) and don't need to be watered during the summer.

I have a deciduous azalea under all day dappled high oak trees... fairly dry. It flowers pretty well, with big rhodo sized bright orange flowers. Looks like this attached pic (not mine). I may be able to find the tag if you were interested.

Here is a link that might be useful: orange azalea pic

Thank you for all the suggestions! I look forward to working some of these into my plans for next year.