Imagine there is a test mass in the vicinity of spinning black hole, but the test mass is kept in place, i.e. it is not in free fall or orbit. Does it experience frame dragging?
My guess would be 'no', as the Lense–Thirring force should be proportional to the velocity of the test mass, in analogy to the Lorentz force. (?)
The Kerr metric has an explicit $g_{\phi t}$ component. From this component, the timelike killing vector of the spacetime, which is the "direction" of the timelike geometric symmetry, inherits a $\phi$ component from here.
If you attempt to keep an object at a constant $\phi$ coordinate while it evolves forward in time, it will not respect the time evolution symmetry, and this will show up as a net force on the object that makes it want to "orbit" along with the direction of the "frame".
A general metric tensor for a spinning physical object will be more complicated than the Kerr metric, but you will still have this same general picture emerge.
