"I'm planning a three week water fast. I haven't detoxified with juices or vegetables at all, I just plan on beginning from where I stand now. Is this dangerous?"
It depends, it could be dangerous if you already are underweight for example, there can also be other reasons why such "diet" may not be recommended.
I advise people that want to do a several weeks water fast to consult their MD or nutritionist before (remember that you are also free to stop the fast at any time should you experience health related issues).
"Also, is it dangerous for me to run while fasting?"
Again it depends, but the answer is likely No. I have recently done a 31-days water fast, and was exercising several hours a day.
Remember to stay hydrated, in fact, you should drink more water during a fast that what you usually drink.
Also, please remember that too much water is harmful to the body; so simply estimate the amount of fluid you currently inject in your body and drink about 1.1 to 1.3 times as much during the fast.
I currently drink a little over a gallon of water every day (and I'm currently not fasting). But I still exercise what people would consider "a lot".