As a counterpoint, on my site (Obsidian Portal), support is a big drain (both timewise and mentally). Further, it has little to do with paid vs free (we're freemium). Both groups demand support, and the freebies are often the most angry.

For example, every day I'm dealing with someone who is convinced that the login system is broken. It's always user error, but it requires input from me pretty much every time, to the point of "here is a new temporary password. i've tested it and it works. please copy and paste it in"

Yes, we have an automated "forgot password" feature, but no matter how much automation you add there will always be people who avoid all that and demand personal attention.