The mother of my 12yr old daughter basically destroyed my relationship with my daughter. Out of jealousy and hatred she told my daughter lies about me. Now my daughter does not want to see me or talk to me. This has been going on for about 5yrs. My bquestion is" If I give up parental rights, do I still have to pay child support to a daughter who wants nothing to do with me?:

Yes.

Giving up your rights, does not eliminate your responsibility to support your child.

First it is doubtful that you could even "give up your rights" but if the mother agreed and the court would agree, your obligation to pay child support is still there, one has nothing to do with the other.

BUT, if the mother has a new partner who wants to adopt, then and normally only then can you give up your rights and have the child support stop.

What you should have done was to go back to court and back to court when they were doing this, and visiting all the time so the child would know what was lies and what was not

I agree... would have been helpful to fight it if you could have, but it is not too late. Your daughter it getting to an age where she will start to make up her own mind about what it true and what isn't. Continue to take the high road, let her be aware that her mother may have said something's out of anger that were not entirely true, and that you would like the opportunity to show her otherwise. Set an example, do not bad mouth her mother in anyway, but simply do what you can to show her with your actions and words that what has been said in the past is not accurate.
If you can e-mail her, do so... if you can send a letter, do that... phone calls, talking face to face if possible, keep the contact pleasant and frequent. Do not back off or this will only make her believe that what her mother has been saying is true. Don't expect her to make the first moves and initiate contact, it will all have to come from you for awhile. Expect to be frustrated, angry, and hurt at times.
It may not happen quickly, but it will happen... even if it takes her getting a bit older and seeing it for herself.