Heck, I oil mine with the strings on it most of the time. It might not be best if you're into that crisp stainless roundwound tone, and you might want to let it dry a little. I'm no expert, but I would apply the oil and give it 5-10 minutes to soak in. At that point, I'd give it a wipe down to pick up anything left on the surface and wait another five to re-string.
I think it is mostly an issue if you have to heavily oil the board because you only do it once or twice a decade. Something on the order of 5-6 drops does my whole board about quarterly, so there's not much slop to get on the strings. Love that lemon oil smell!