Yes, you'd set up a Donor Advised Fund, transfer the stock to that fund, and sell it from within the fund, or donate the stock directly to an large established nonprofit that is capable of accepting donations of stock.
I don't think that's what the person is asking. He's asking if you can avoid ALL taxes on RSUs by donating all of it. And the answer is NO.
You can't transfer an RSU grant and when it vests, the taxes are automatically paid for by the company (RSU manager) selling 33% (varies) of the shares.
Once you have the shares you can do the above and can avoid taxes on the gains of post-tax shares.
