It really depends on the specific chain lube. Some (like one I won't name so I'm not accused of shilling) do a great job keeping water out of and off the chain so you can let it drip dry. Others, especially some of the very dry, and various wax lubes don't do such a great job and you risk rust.

Minor rust where you can see it on the outside of the chain doesn't really matter, but if there's water in the chain rust can cost you some chain life. One incident probably isn't a big deal anyway, but if you keep letting rust attack the chain, you'll both see short lift, and if neglected totally you could crack a plate, and as anyone whose ever broken one, that's not fun.