Nothing against anyone in this forum, but I am leery of proprietary databases...
I am wondering if there are any viable open-source replacements for MySQL now that Oracle owns it.
It seems like the bigger problem with any open-source database is finding an active support community which have all but dried up online. 
All but dried up seems a bit dramatic?
If you want non proprietary db is to write it yourself ( exaggeration) 
But I think there is more to your post than meets the eye. What challenges are you facing and how can we help you?
I said that online support (e.g. active online forums) are all but dried up.
If you can't see that, then you haven't been online very long. 
I have been away for a few years and now that I am back tinkering with IT all of my favorite online user forums are dead.
Never would have thought that would happen.
This seems to be an active database forum, and that is great, but it is one of the few that has survived the times.
1.) I don't want to use a proprietary database for my startup because I cannot afford one.
2.) MySQL was a great open-source database before Oracle took over it, and rumors started of its ultimate demise.
3.) I was interested in trying PostgreSQL, but the forums for it haven't been updated in years.
THAT is my problem...
I'm not a genius like many of you, and I need an active online community to help me along with whatever open-source database I choose.
Make sense?
What is online? 
I dont consider myself a genius either just hacking away baby.
Have you also looked at MongoDb?
MongoDB isn't a serious database.
A lot of people mention MariaDB as it forked from MySQL, but I'm not sure that it is guaranteed to be around for long either.
Back in the day, PostgreSQL was considered an enterprise open-source database, and it had an active online community, but it looks like any forums have went offline.
I think MariaDB is what you're looking for. A LOT of people have a big investment in MySQL. Those that have licensing concerns will eventually move to MariaDB. I think it will be around a long, long time.
Do you understand how MySQL (under Oracle) licensing works now?
And do you understand how it might change in the near future?
Maybe I am fearful of something that I really do not understand? 
Here is what I don't want to happen...
I spent a whole bunch of time learning MySQl and building a hunormous database, get my startup business going, and then find out that I own Oracle tens of thousands of dollars to legally run my website and keep it going.
(I have always be rather clueless on what the "fine print" of End-User License Agreements...)
Can you, or anyone else, help me to better understand things from a legal standpoint of what the implications are or could be?
Maybe I am worrying about things that do not apply?
The challenge is that you're on a message board for Microsoft SQL Server and not MySQL or one of the other open source databases. I know very little about MySQL licensing -- now or in the future. 
If those are your concerns, I would look first at MariaDB instead of MySQL. It is trying to solve the exact problem you are describing. You should be able to find some good articles on their license.
I think the larger problem is that it is hard to find active user forums anymore on most IT topics which is why I am here.
I only see one semi-active forum for MariaDB and while there are a few posts this month, most are from last year.