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.
My family has preserved olives for years. Yes, perfectly good olives fizz. We preserved ours with salt and a small amount of vinegar, then filled the quart jar with water, covered the olives and allowed to rest for months. This causes fermentation and the olives will fizz. I have also had this happen with green olives I preserved in a brine (no vinegar), and this causes olives to fizz as green olives are also acidic. They are delicious and perfectly edible. Locally preserved olives are most likely fermented, but to be safe check with the producer.