There are some fanatstic orange begonias. I got mine at Sunnyside nursery in Tsawwassen. Also, New Guinea impatiens come in orangey shades.
What about the Aquilegia formosa in this picture: http://www.botanicalgarden.ubc.ca/forums/showthread.php?s=&threadid=570 ? There's a heuchera that has reddish-orange flowers. They may be more red than you want, though - my colour memory isn't that good. Or some of the astilbes (particularly after the flowers have dried!). I've never bought one that had the advertised flower colour though.
Annuals-Orange Tuberous Begonias, Fibrous begonias can have bronze foliage,Coleus,Pansies,Abutilon Perennials--Astilbe, bronze foliage,
In regards to vines, a nice orange flower is Orange Clock Vine (Thunbergia gibsonii) and also Orange Honeysuckle (Lonicera ciliosa). :)