It's hard to say anything about "flux" in general because there are so many different types of it out there.
For example, SparkFun sells (out of stock right now) a Kester 2331-zx flux pen. It's water-soluble so water should clean it off, but it's organic acid, so it's corrosive over a long period of time if you don't clean it. Here's an example of what happened when someone put that flux on a circuit board and let it sit for a month or so. The advantage of this flux is I believe it's more active than other types of flux, but you just have to make sure to clean it off when you're finished. Aside from that first link which also discussed conductivity, I've heard of other people claiming to have experienced random problems that could seemingly be attributed to water soluble flux having slight conductivity, or at least some kind of property that interfered with a circuit. But either way, that should only apply if it's not cleaned off.
On the other hand, there are also plenty of no-clean fluxes out there. Those should be safe to leave on the circuit board after you're done. I personally prefer a Kester 186 pen. As Kester itself says, it's a rosin flux that leaves behind non-corrosive and non-conductive residue. (I clean it off anyway, though--it's ugly and sticky)
Bottom line: all fluxes aren't created equal. Use a good no-clean flux and you should be in good shape
For example, SparkFun sells (out of stock right now) a Kester 2331-zx flux pen. It's water-soluble so water should clean it off, but it's organic acid, so it's corrosive over a long period of time if you don't clean it. Here's an example of what happened when someone put that flux on a circuit board and let it sit for a month or so. The advantage of this flux is I believe it's more active than other types of flux, but you just have to make sure to clean it off when you're finished. Aside from that first link which also discussed conductivity, I've heard of other people claiming to have experienced random problems that could seemingly be attributed to water soluble flux having slight conductivity, or at least some kind of property that interfered with a circuit. But either way, that should only apply if it's not cleaned off.
On the other hand, there are also plenty of no-clean fluxes out there. Those should be safe to leave on the circuit board after you're done. I personally prefer a Kester 186 pen. As Kester itself says, it's a rosin flux that leaves behind non-corrosive and non-conductive residue. (I clean it off anyway, though--it's ugly and sticky)
Bottom line: all fluxes aren't created equal. Use a good no-clean flux and you should be in good shape