Q:

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective? 

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules into the dual of tensor product.  My question is, why is this map always injective?  

A:

This is false.   For instance, let $k$ be a field and let $R=k[x_0,x_1,x_2,\dots]/(x_0,x_1,x_2,\dots)^2$.  Let $V=W$ be a free $R$-module with basis $\{e_0,e_1,\dots\}$ and let $\alpha\in V^\vee$ be given by $\alpha(e_i)=x_i$.  Note that $\alpha\otimes\alpha$ maps to $0$ in $(V\otimes_R V)^\vee$, since $\alpha(e_i)\alpha(e_j)=x_ix_j=0$ for any $i,j$.  However, I claim $\alpha\otimes\alpha$ is nonzero in $V^\vee\otimes_R V^\vee$.
To prove this, note that $R$ is the direct limit of the subrings $R_n=k[x_0,\dots,x_n]/(x_0,\dots,x_n)^2$ and so $V^\vee\otimes_R V^\vee$ is the direct limit of the tensor products $T_n=V^\vee\otimes_{R_n} V^\vee$ over $R_n$.  It thus suffices to show that $\alpha\otimes\alpha$ is nonzero in $T_n$ for all $n$.
Note that $V^\vee\cong R^\mathbb{N}$, with the coordinates corresponding to evaluation at the $e_i$.  As an $R_n$-module, $R$ splits as a direct sum $R_n\oplus k^{\oplus\mathbb{N}}$ where $k$ is an $R_n$-module by letting the variables act trivially and the second summand is the vector space spanned by the $x_i$ for $i>n$.  Thus as an $R_n$-module, $V^\vee\cong R_n^{\mathbb{N}}\oplus (k^{\oplus\mathbb{N}})^{\mathbb{N}}$; let us call these two summands $M$ and $N$.  Now notice that $N$ is annihilated by the maximal ideal of $R_n$ and so $N\otimes_{R_n} N$ can be identified with $N\otimes_k N$.  But when you take a tensor product over a field, the tensor of any two nonzero vectors is nonzero.   In particular, since the projection of $\alpha$ onto $N$ is nonzero, so is the projection of $\alpha\otimes \alpha\in T_n$ onto $N\otimes_{R_n} N$.  Thus $\alpha\otimes\alpha$ is nonzero in $T_n$, as desired. 

A:

No, this is false.  Even if $V$ and $W$ are free $R$-modules, this can be false.  Indeed, let $N$ be the set of natural numbers, and let me use the notation $M^*$ for what you call $M^\vee$. Then, Goodearl (in the second-to-last sentence of K. R. Goodearl, Distributing tensor product over direct product, Pacific J. Math.  43, Number 1 (1972), 107--110) constructs a commutative ring $R$ such that the canonical map $R^N \otimes R^N \to R^{N\times N}$ is not injective.  In view of the canonical isomorphisms $R^N \cong \left(R^{\left(N\right)}\right)^*$ and $R^{N\times N} \cong \left(R^{\left(N \times N\right)}\right)^* \cong \left(R^{\left(N\right)} \otimes R^{\left(N\right)}\right)^*$, this rewrites as follows: The canonical map $\left(R^{\left(N\right)}\right)^* \otimes \left(R^{\left(N\right)}\right)^* \to \left(R^{\left(N\right)} \otimes R^{\left(N\right)}\right)^*$ is not injective.