Like the title says, I opened a jar of locally produced olives and it started to bubble like seltzer. Is it unsafe to eat?
Yes, throw it away.
Bubbling, fizzing, pressure etc. indicates some kind of microbiological activity that is unwanted for properly canned food - canning should eliminate these organisms. Any behaviour like the one you described indicates that something went wrong, so you can not assume the contents of the jar or can to be safe.
Not necessarily. It depends of whether it is expected or not. Some curing methods result in this. I was offered such olives once. They were kept in a big pressured plastic soda bottle.
They were delicious, I ate all of them and I was perfectly fine.
The person who gave them to me rightfully warned me to unscrew the cap slowly and vent the pressure gradually.
However I would be surprised to see any commercial (supermarket) products prepared in this manner.
You say locally produced, so your best bet is to ask the producer. If you don't trust them then you should not consume any of their produce anyway.
Yes, the olives should be thrown out. They might look good and smell good but taste bad.