When sitting inside a boat the action of using bow no longer results in an arrow being fired. The motion and sounds of the firing action are the same but nothing comes out. I have tried multiple combinations and the result is the same. Even flaming arrows, which should leave some evidence of having been fired when they make contact with the ground or a mob, completely disappear.
Try shooting it straight up and see if that works. The arrows might be weirdly hitting the hitbox of the boat but not doing anything to it.
Confirmed. I just head out on smp2 to try this and for sure: I'm aiming way above any hitboxes and should be able to hit land here. Note: my arrows are flaming too (voters bow) so even if I don't hit anything we should clearly see the arrow. Well.... Nothing at all. This seems like a bug to me as well, though in all honesty: I haven't tried this out in single player just yet. Will do that later on, if that changes things I'll post another reply, otherwise I'll just update my post.
It looks like this issue has already been reported and remains unresolved at this time. https://bugs.mojang.com/browse/MC-51126?jql=project = MC AND text ~ "bow boat"