There is no non profit organizations that want a timeshare or want to deal with disposing of a timeshare. There is at least one organization that you can pay several thousand dollars, then they put the timeshare on ebay and will donate some or all of what it actually sells for to a non profit.
If your timeshare is one that will sell easily for $5000+ they may take it at no cost to you and then donate a small portion of that to a charity.
If you want to help a charity donate to it directly and deal with disposing of your timeshare yourself. Many timeshares have no value. If you think your has value you can sell on TuG marketplace, Redweek, Craigslist or ebay. If you do some research and discover it has no value or very little value, you can give it away on the bargain forum. You need to have all the details about the unit. If you can't find a taker you may need to add incentive like paying closing costs and paying 2016 MF but giving 2016 usage to the new owner.