SMTP authentication is required.

You have enabled SMTP authentication for the IP range that the user is connecting from, but the user has not configured his client to use SMTP authentication. There's two ways to solve this problem. Either configure your email client to use SMTP authentication. This setting is normally found in the account settings in your email client. Or, disable SMTP authentication for the IP range. The first solution is recommended since it reduces the risk that anyone will send spam through your server.

By default, hMailServer does not require SMTP authentication for connections coming from localhost / 127.0.0.1. For connections coming from other hosts, SMTP authentication is required for deliveries to external recipients. By default, hMailServer never requires SMTP authentication for deliveries to local accounts, since that would prevent other e-mail servers to deliver email to your installation.

Wow. Quick response time, I'm impressed.
*adds a mark to the "Send hMail people some money" column.

I forgot that SMTP authentication is handled it two places. I went back and checked the "IP RANGES" section, and noticed that the "external address delivery" box had been checked.

I unchecked it, and all seems to be working again.