Some functions of non-green flower colors:
#1: plant-animal interaction. Flower colors act as signals received by animals with eyes (e.g., pollinators, which many plants need to attract in order to reproduce).
However, it sounds from your question as if you're looking for other physiological effects. Two that come to mind are:
#2: heat. Flower color and opacity have a great effect on flower temperature.
This can be especially important for arctic and alpine flowers. Many of these are not only shaped like satellite dishes, but even pivot to follow the sun. This creates a warmer micro-environment which can be important to attracting pollinators and to increasing/speeding seed production. (See, e.g. Mølgaard 1989)
#3: UV protection. Certain pigments can be more effective at filtering UV.
I work in the alpine and occasionally encounter "glasshouse" plants, which have translucent floral parts that act to create warmer microenvironments, with the same effects as above. In addition, these floral parts have been shown to be more effective at UV filtering than green floral parts (Tsukaya 2002)
*edit: as these papers are behind paywalls, I'll also point out that the Wikipedia pages on Heliotropism and Rheum nobile contain similar information.