I don't think I would ever do it, but I'm sure there is some safe dye that would work. I knew a Bichon named El Gringo who had the top of his head dyed purple most of his life. It was his "signature" and everyone loved it. His owner would never have done anything to harm him and he was fine with it. Bichons and white Poodles would probably be easy to dye.
Oh yeah you totaly can do it, I've done it for halloween to a pooch. The secret? KOOLAID! here's how to do it How to Dye Hair with Kool Aid | eHow.com completely 100% safe, and if they want to taste their hair all the power to themI've never seen a rinse but I know they exist I see nothing wrong with dying a pups hair for a purpose, we dye our's with actual chemicals, i'm not sure what the rinses are made of tho...
That being said, I remember watching a dog whisperer episode where a woman was completely obsessed with the colour pink to the point where she permanently dyed her bichon pink. I thought it was complete overkill, but the dog was well cared for and loved. I think it might be a 'his or her thing' although I admit I would never constantly dye bijoux yellow because I like the colour, but I do see nothing wrong with it as long as the dog isn't suffering in anyway.
This is what I did to dye the hair without conditioner
Using Drink Mix
1)Choose an unsweetened powdered drink mix that results in pink liquid, such as strawberry. Empty the packets into a large spray bottle. The number of packets you will need depends on the size of the poodle and the shade of pink you desire.
2)Add warm water to the drink mix in the spray bottle and shake it up to dissolve. Add the amount of water that gives you the shade of pink you desire. More water will give you a paler pink, while less water will give you a more intense pink.
3)Wash your dog and rinse it thoroughly.
4)Spray the dissolved drink mix and water onto your poodle's fur in small sections, working the dye into the fur.
5)Start at the back of the dog and work your way forward until you have completely dyed the dog. Use car when working around its eyes.
Read more: How to Dye a Poodle Pink | eHow.com How to Dye a Poodle Pink | eHow.com