Here is roughly the philosophy of the Weitzenbock technique. (Most of what follows is taken from Berline-Getzler-Vergne book.)
Suppose that $E_0,E_1\to M$ are vector bundles on an oriented Riemann manifolds $M$ equipped with hermitian metrics. Denote by $C^\infty(E_i)$ the space of smooth sections of $E_i$.
A symmetric 2nd order differential operator $L: C^\infty(E_0)\to C^\infty(E_0)$ is called a generalized Laplacian on $E_0$ if its principal symbol $\sigma_L$ coincides with the principal symbol of a Laplacian. Concretely this means the following.
For a smooth function $f\in C^\infty(M)$ denote by $M_f$ the linear operator $C^\infty(E_0)\to C^\infty( E_0)$ defined by the multiplication with $f$. Then $L$ is a generalized Laplacian if for any $f_0,f_1\in C^\infty(M)$ and any $u\in C^\infty(E_0)$ we have
$$ [\; [\; L,M_{f_0}\; ], M_{f_1}\; ]u = -2g( df_0,df_1)\cdot u $$
where $[-,-]$ denotes the commutator of two operators. Equivalently, this means
$$[[L,M_{f_0}],M_{f_1}]=-2M_{g(df_0,df_1)}. $$
One can show that if $L$ is a generalized Laplacian on $E_0$, then there exists a connection $\nabla$ on $E_0$, compatible with the metric on $E_0$, and a symmetric endomorphism $W$ of $E_0$ such that
$$ L =\nabla^*\nabla +W. $$
The classical Weitzenbock formulas give explicit descriptions to the Weitzenbock remainder $W$ and the connection $\nabla$.
Usually the generalized Laplacians are obtained through Dirac type operators which are first order differential operators $D: C^\infty(E_0)\to C^\infty(E_1)$ such that both operators $D^\ast D$ and $D D^\ast$ are generalized Laplacians on $E_0$ and respectively $E_1$. We can rewrite this in a compact form by using the operator
$$\mathscr{D}: C^\infty(E_0)\oplus C^\infty(E_1)\to C^\infty(E_0)\oplus C^\infty(E_1), $$
$$\mathscr{D}(u_0\oplus u_1)= (D^*u_1)\oplus (D u_0). $$
Then $D$ is Dirac type iff $\mathscr{D}^2$ is a generalized Laplacian.
The Weitzenbock remainders of $D^\ast D$ and $D D^\ast$ involve curvature terms. If the Weitzenbock remainder of $D^*D$ happens to be a positive endomorphism of $E_0$, then one can conclude that
$$\ker D=\ker D^\ast D=0. $$
The Hodge-Dolbeault operator
$$\frac{1}{\sqrt{2}}(\bar{\partial}+\bar{\partial}^*): \Omega^{0,even}(M)\to \Omega^{0,odd}(M) $$
on a Kahler manifold $M$ is a Dirac type operator. For more details and examples you can check Sec. 10.1 and Chap 11 of my lecture notes.
Best Answer
I don't know of a proof that would really be characterized as global, and if I saw one I would immediately try to figure out how it's really local. There is, however a proof along different lines than that in GH or Voison, and seems more enlightening to me. Huybrechts in his Complex Geometry book gives one that is more representation theoretic/linear algebraic. First he proves a formula (due to Weil?): for a primitive k-form $\alpha \in P^k$ we have:
$\ast L^j (\alpha) = (-1)^\frac{k(k+1)}{2} \frac{j!}{(n-k-j)!} L^{n-k-j}I(\alpha)$
Here n is of course the complex dimension of the manifold, and $I$ is the operator induced on forms by the almost complex structure (page 37 of Huybrechts). This underappreciated formula seems to only be found in this book.
He then uses this together with the purely linear algebraic Lefschetz decomposition on $\alpha$ and $d\alpha$ to prove the commutation relation in the form $[\Lambda, d] = -d^c$. The proof is a bit calculational though. See page 121-122.