Depends on the event. Offhand, figure, at a minimum, you're looking an investment of $5-7mm per year for FedEx Cup events, less for Fall Series events. WGC events are more.
This is current "market" prices. For a tournament that lost a sponsor, figure that number is subantially less (i.e. the former Wachovia tournament in Charlotte). Plus, with tv deals set to come up for renegotiation in a couple years, you may well see a "sea change" of sorts.
Of that amount, you're also doing ad buys throughout the season, with a heavy ad buy during the event you sponsor. You're also paying for signage, and other expenses during the event. You get some slots in the Wednesday pro-am, your name all over the place at the event, and your head honco gets to go on network tv and sound like a muppet while talking about what a great event it's been.
The tournament itself pays the purse, however the tournament gets a big chunk of the sponsorship revenues. The course (or body managing course) hosting the tournament gets a fee (or a percentage of profits). For a private course, it's typically a fee to offset the fact that their members lose their course to the Tour for 2-3 weeks. If the PGA Tour is using one of their own (i.e. TPC Sawgrass), then it's a sweetheart deal for the Tour.
If not all, then almost all of the organisations running events are 501(c)3 organisations (i.e. nonprofits).