My tonsils are swollen most of the time. My teacher says to open the back of my throat so I do, but it doesn't work . I don't know what I'm doing wrong. Can it affect the tone? Any other suggestions on how to get a rich tone?

Wow. I hope you're okay. As far as I know, big tonsils affect breathing, so that should affect your tone!

Try saying 'O' from the back of your throat. If you can do it and make a better tone, then it might be your lips/teeth/embouchure making the poor flute tone.

I find my lips make more difference to tone ...to my playing that is .., than the throat...

This is a condition that could be caused by infection and could develop into chronic state. I suppose you have consulted a doctor, if you have not, I suggest you do. Some informations here:
http://www.itonsil.com/tonsils_tonsillitis.html