Nope, not built-in. One could rig it with expressions and Bezier Warp to some degree, though. Another way might be to use the puppet tool on a pre-comp - animate the arrow, use its path as a path e.g. for the Stroke effect in the pre-comp. Then in the parent comp let the Puppet tool generate its mesh based on the dummy stroke. turn of the stroke and reveal tghe arrow in the pre-comp. If there is enough detail in teh puppet mesh, it should reasonably fo9llow the path. Same technique would work with otehr distortion effects also...
Depending on the complexity of the path and the arrow you could make this work with just a couple of shape layers. One for the arrow head, the other for the feathers. Tie the tail to the head's position layer by applying the following expression to the tail layer position property:
p = thisComp.layer("Head").transform.position; np = p.valueAtTime(time - .3)
The anchor point of the head or point layer should be at the back center of the arrow, the anchor point of the tail would be at the front of the feathers. Then add a path to a solid, apply stroke to the path, copy the path and select the head layer's position property and paste. Head and Tail layers are set to Auto Orient to Paht.
Now simply adjust the start and end points of the path to make the center of the arrow. It would look something like this:
You'll find a CS5 project file HERE
You'll notice from the video that I failed to trim the comp to hide the first .3 seconds where the tail isn't moving...