I'm very closely involved with a non-profit institute, which has charity status by the IRS (contribs to it are tax-deductible). Is there a way to get the money from a 403(b) contributed to these organizations upon my (distant future) death free of the income-taxes that normally occur when you do withdrawals?

Name them as the beneficaries of your retirement benefits. All charitable contributions are exempt from estate taxation. There is no income tax on a transfer after death.

Great! Thanks for the helpful info. Now I can plan accordingly (namely, if I need to use the money in my retirement accounts, I'll use my 403b money last, to possibly avoid being taxed on it at all).