If just a small jar, okay to keep on the counter for a few weeks after opening but check for funky stuff floating on top. Like maybe not more than 2 or 3 weeks and definitely the weather will affect this. In winter, probably ok for longer but if it's summer and hot it will go bad much faster.

A larger container should be refrigerated after opening though, as all the peeps here have so eloquently stated. Nothing worse than opening a nice big opaque container of maple syrup, pouring it out on some nice fresh pancakes, and discovering big glops of mold have landed on your breakfast!