Even if it's not technically paleo, I'm wondering what place, if any, it has in a healthy diet. Personally, I love some on my avocado :)
Per Cordain, no it is not strictly paleo. Nothing related to it being "healthy", just the impact on acid-base balance. That being said, many paleo Gurus such as Rob Wolff don't worry about it. Most folks, myself included, use vinegar on salads and in sauces.
I have found that the sugar in really good balsamic vinegar triggers huge food cravings. I was eating delicious salads with a balsamic & olive oil dressing alongside my meat and then wondering why I suddenly became incapable of maintaining any IF the following day. Now that I've cut it out I feel much more stable, don't need to run to the fridge every two minutes and able to IF. But then I am super-sensitive to all things non-paleo.
Total anecdata, but I think I've identified balsamic vinegar as one of the things that can really help with constipation. I know some people recommend apple cider vinegar in water - I just put balsamic on my salad and it helps the same way.