Are dual bases and the Hodge dual "entirely distinct" uses of the word "dual", as per MTW? NB: Basis one-forms and contravariant basis vectors (which, following Menzel, I am calling reciprocal) are the same thing. See, for example, the Mathematical Appendix to Gravitation and Inertia, by Wheeler and Ciufolini.
MTW, in Exercise 3.14. DUALS, we are told 

[A previous and entirely distinct use of the word "dual" (section 2.7) called a set of basis one-forms $\left\{ \omega^{\alpha}\right\} $ dual to a set
  of basis vectors $\left\{ \mathbf{e}_{\alpha}\right\} $ if $\left\langle \omega^{\alpha},\mathbf{e}_{\beta}\right\rangle =\delta^{\alpha}{}_{\beta}.$
  Fortunately there are no grounds for confusion between the two types
  of duality. One relates sets of vectors to sets of one-forms. The
  other relates antisymmetric tensors of rank $p$ to antisymmetric
  tensors of rank $4-p$.]

I'm not so sure these uses of "dual" actually are "entirely distinct". Consider the definition given by Menzel (Mathematical Physics) for the reciprocal (contravariant) basis for a 3-dimensional curvilinear coordinate system in Galilean 3-space.
Rectangular Cartesian coordinates are written $x^{i},$ with their
primary basis vectors written $\hat{\mathfrak{e}}_{i},$
and the corresponding reciprocal basis vectors are written$\hat{\mathfrak{e}}^{i}$.
For curvilinear coordinates $q^{\bar{i}},\mathfrak{e}_{\bar{i}}\equiv\hat{\mathfrak{e}}_{i}\frac{\partial x^{i}}{\partial q^{\bar{i}}},\mathfrak{e}^{\bar{i}}$
will be used. The symbol $\mathcal{E}_{\bar{i}\bar{j}\bar{k}}$ denotes
a tensor density of weight $+1$, and is identical to the Levi-Civita
tensor ($\varepsilon_{ijk}$) in orthonormal coordinates. Following Menzel we define the volume element $V$ and the reciprocal basis vectors $\mathfrak{e}^{\bar{i}}$ as follows:
$$
V\equiv\mathfrak{e}_{\bar{1}}\cdot\mathfrak{e}_{\bar{2}}\times\mathfrak{e}_{\bar{3}},
$$
$$
\mathfrak{e}^{\bar{i}}\equiv\frac{\mathfrak{e}_{\bar{2}}\times\mathfrak{e}_{\bar{3}}}{V}.
$$
From standard tensor calculus, we may also express the volume element
as the Jacobean determinant of the transformation matrix:
$$
\mathcal{E}_{ijk}\frac{\partial x^{i}}{\partial q^{\bar{i}}}\frac{\partial x^{j}}{\partial q^{\bar{j}}}\frac{\partial x^{k}}{\partial q^{\bar{k}}}=V\mathcal{E}_{\bar{i}\bar{j}\bar{k}}=\varepsilon_{\bar{i}\bar{j}\bar{k}}.
$$
We now express the cross product of two barred basis vectors on the
unbarred basis, and use the conventional transformation method to
obtain the following:
$$
\mathfrak{e}_{\bar{2}}\times\mathfrak{e}_{\bar{3}}=\mathcal{E}_{ijk}\frac{\partial x^{j}}{\partial q^{\bar{2}}}\frac{\partial x^{k}}{\partial q^{\bar{3}}}\hat{\mathfrak{e}}^{i}
$$
$$
=\mathcal{E}_{ijk}\frac{\partial x^{j}}{\partial q^{\bar{2}}}\frac{\partial x^{k}}{\partial q^{\bar{3}}}\frac{\partial x^{i}}{\partial q^{\bar{i}}}\mathfrak{e}^{\bar{i}}
$$
$$
=V\mathcal{E}_{\bar{1}\bar{2}\bar{3}}\mathfrak{e}^{\bar{1}}
$$
Which gives us back Menzel's definition. More generally we have
$$
\mathfrak{e}_{\bar{j}}\times\mathfrak{e}_{\bar{k}}=V\mathcal{E}_{\bar{i}\bar{j}\bar{k}}\mathfrak{e}^{\bar{i}}=\varepsilon_{\bar{i}\bar{j}\bar{k}}\mathfrak{e}^{\bar{i}},
$$
which is formally very similar to the first equation $\ast J_{\alpha\beta\gamma}=J^{\mu}\varepsilon_{\mu\alpha\beta\gamma}$
of (3.51) in MTW, Exercise 3.14.
If we replace the cross product with the wedge product, it seems that we could extend this definition of reciprocity to Minkowski 4-space.  So, am I correct in concluding that the two uses of the term "dual" are not actually "entirely distinct," and in fact are closely related?
 A: In three Euclidean dimensions basis duality and Hodge duality are identical.  When a fourth dimension is introduced the identity bifurcates.  One aspect of the duality becomes the identification of a volume 3-form (in 4-space) with a vector.  The other is the condition $\mathbf{d}x^{\mu}\mathbf{e}_{\nu}=\delta^{\mu}_{\nu}.$
This is my rendering of Menzel's development for the case of Euclidean 3-space on rectangular Cartesian $x^i$ and general cruvilinear $q^i$ coordinates.  The appearance of two basis vectors side-by-side is dyadic notation.  Replace the cross with a wedge.  The over-hat $\hat{\mathfrak{e}}_i$ indicates an orthonormal basis.  All others are assumed general.

A: MTW (Gravitation by Misner, Thorne, and Wheeler) sec 2.7 makes no use of the volume element, while MTW sec 3.5 (Ex 3.14) does make use of the volume element.
The duality in MTW 2.7 is akin to the duality between points and [hyper]planes, column vectors and row vectors.
However, the Hodge-duality in MTW 3.5 (ex 3.14) is a different pairing using the volume element.
It relates a column vector to an antisymmetric matrix, which have the same number of components only in 3 dimensions. The use of the volume element
is associated with the notion of pseudo-quantities (like a pseudovector).
(I'm not going to comment about Menzel since the question in the title is about MTW
and I would have to decode those sections in Menzel.)
Possibly useful resources:

*

*https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf (Ch 27 is on Hodge)

*Applied Differential Geometry by William L. Burke

*Tensor Analysis for Physicists  by J. A. Schouten

