I know Ted Turner did it with the UN Foundation, although that isn't technically the UN itself.
Are there other examples?
They certainly can. I don't how many people actually do it, but most governments will accept direct cash donations. E.g. here's a page on how to make a gift to the US Government: https://www.fiscal.treasury.gov/public/gifts-to-government.h...
Edit to add: Here's some data on donations made to reduce the federal debt https://www.treasurydirect.gov/govt/reports/pd/gift/gift.htm . You can see that this seems to average a few million dollars per year.