It's the other way around. It's the non-profit part of the donations to non-profits that relieves the non-profit from paying taxes on the revenue, not the fact

It's the other way around. It's the non-profit part of the donations to non-profits that relieves the non-profit from paying taxes on the revenue, not the fact that it's a donation. If you somehow made a voluntary donation to, say, your local bar, the bar would still be taxed on that revenue (assuming it's not somehow a registered non-profit bar).