Does Mach's principle imply that the gravitational field has a non-zero curl? I would normally visualize the gravitational field as "radial", i.e., one whose curl is zero. However, while thinking about Mach's principle, particularly the notion of frame-dragging (as illustrated in this video at min 6:43), it seems that the curl has to be non-zero to account for experiments as mundane as the rotating bucket of water.
Would this mean in turn that an object dropped from rest towards a massive, rotating star wouldn't follow the radial direction (i.e., shortest path) but would actually be dragged slightly in the direction of rotation as if going down a whirlpool?
PS: Please keep in mind that I only have a basic, qualitative understanding of General Relativity.
 A: Indeed, a massive rotating mass, will induced a n angular motion to freely falling objects. A ballerina falling freely towards the rotating mass will get a rotational motion when stretching her arm in the radial direction. The longer her arms the faster she spins, which is quite the contrary to a spinning ballerina stretching her arms on the dance floor, whose angular momentum has to be conserved.
A spinning ballerina in an empty space will feel an outward force pulling on her arms. What force causes this? Is there some weird kind of a gravity field? Well, just like an accelerated observer in empty space can find out he/she is actually accelerated (by measuring tidal effects and looking at clocks), so can the ballerina do that too.
The spinning bucket can spin in a flat spacetime without rotation (only a real mass spinning around her, like a spherical massive shell, can induce rotation by gravity). A heavy spinning mass gives spacetime a rotation component.
A: You're talking about three things here: (1) Mach's principle, (2) frame dragging, and (3) the gravitational field.
2 is an effect in general relativity. 3 is something that exists in Newtonian gravity but not in GR. 1 is an idea that is somewhat embodied in GR but not 100%.
Newtonian mechanics has a 3-vector field called the gravitational field. Newtonian mechanics doesn't embody Mach's principle, so you can't use Mach's principle to infer anything about the gravitational field.
GR doesn't have a 3-vector field called the gravitational field. Therefore there's no way to discuss within the context of GR whether such a field has a curl.
A: Mach's principle and Newton's bucket
Mach's principle was a philosophically motivating factor for general relativity, but it is not a necessary postulate of GR.  As the first sentence of the Wikipedia page on Mach's principle states, Mach's principle is an imprecise hypothesis.  It is not a formal, mathematically precise statement.
By looking at frame dragging, it makes sense to say something like: "the curl of the gravitational field in GR is not universally zero."  But I would not say this is a direct consequence of Mach's principle.  Mach's principle does not imply frame dragging.  It's perfectly reasonable to talk about Mach's principle in the context of non-relativistic mechanics where there is no frame dragging.
Newton's rotating bucket thought experiment does not require gravity to have a non-zero curl.  Newton formulated that experiment as an argument for the existence of absolute space.  The centrifugal effect is a psuedo-force due to the non-inertial reference frame of the bucket.
I think a GR-centric way of thinking about Newton's bucket, is that the bucket rotates relative to the local gravitational field (which is just spacetime).  The local field/spacetime is  determined by the matter elsewhere, so the Machian statement "matter elsewhere influences which frames are inertial here" is true.
frame dragging and the curl of $\vec{g}$
In classical Newtonian gravity the gravitational field is curl free.  There is no frame dragging in classical gravity.  A rotating point mass produces the exact same gravitational field as a non-rotating point mass.
General relativity has frame dragging.  The rotating point mass produces a different field than the non-rotating one.  This is evidenced by the fact that the spacetime around a rotating and non-rotating black hole are described by different metrics, the Kerr and Schwarzschild, respectively.
The curl operator is only defined for 3-vectors, as is the vector cross product.  So it doesn't make sense to think about the curl of a metric tensor in GR.  However, you can take weak field limits of GR and get back approximate formulations with 3-vectors.  If we find the weak field limit of the Kerr metric for a rotating point mass, we will find that there is a magnetic-like term.  This approximate formulation looks almost exactly like Maxwell's equations. The effects of frame dragging can be approximated to first order as a Lorentz-like force.  This approximate formulation of GR is sometimes called gravitoelectromagnetism.  People sometimes call frame dragging an example of a gravitomagnetic effect.
An object dropped toward a rotating star or black hole will indeed pick up some non-radial motion.  A common calculation for students of GR learning about the Kerr metric is to determine the motion of an object dropped from rest in the rotational plane from $r=\infty$.  The object has a $\ddot{r}$, it speeds up in the radial direction as it falls.  Additionally it has a $\ddot{\phi}$, it gains angular momentum in the same direction as the rotation, moving in the azimuthal direction around the central mass.
If you remember your right hand rules from E&M, you can figure out the same qualitative effect happens when you drop a negatively charged particle towards a rotating positively charged ball.  The central charge has both an electric and a magnetic field.  The electric field causes the attraction and the magnetic field causes the azimuthal motion.
It turns out this whirlpool path is the shortest path or geodesic in the spacetime, because the spacetime is moving.  To fall straight in, the object would need to go "against the flow" of space and travel a greater distance.
Remember, freely falling reference frames are inertial.  The rotating central mass has different inertial frames than a non-rotating mass, where things fall radially.  The Machian statement that "matter elsewhere influences which frames are inertial here" is true.  In the case of frame dragging the motion of the matter elsewhere is important too.  Frame dragging is consistent with Mach's principle, but that doesn't mean Mach's principle implies frame dragging.
