As a followup:
I contacted DreamHost tech support, and they replied that they had just setup a system that blocked any outgoing email that showed a different host in the From: setting. I had been using a gmail address as the forum's "Board Email Address". This had worked fine when initially set up, and had worked for years on other sites, but DreamHost suddenly blocked it without warning anyone.
Their response also said the change had been "rolled back", but I still wasn't getting the email, so either that statement was incorrect, or their rolling back did not really take effect.
I realize what I was doing is a poor practice, but it had worked the past three years. I set up an email account within DreamHost and set that email to automatically forward everything to the gmail account, and all is back working.
However, it was a terrible mistake to move to DreamHost. I opted to use their shared server service, and have had site outage problems and database not available issues like crazy.
I chose DreamHost because another UBB Threads forum runs there, and that admin had good results. Only he is using a VPS or some level above what I started at.