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First, since $\beta_{R,M,N}$ is a natural transformation of additive functors, this is obviously an isomorphism when $N$ is a finite free $R$-module. (Edit. As Jeremy Rickard observes as well, that is probably enough to conclude, since a maximal Cohen-Macaulay module of finite injective dimension appears to be projective.) Second, since this is a local problem, it suffices to prove the case when $R$ is a local, Noetherian, Gorenstein ring of some dimension $d$ (local Noetherian rings have finite Krull dimension). The result is proved by induction on $d$. I will use some results from Chapter 21 of the following. (Aside: This is the second time I have mentioned Chapter 21 on MathOverflow in two days.)

First, since $\beta_{R,M,N}$ is a natural transformation of additive functors, this is obviously an isomorphism when $N$ is a finite free $R$-module. (Edit. This is probably enough to conclude, since a maximal Cohen-Macaulay module of finite injective dimension over a Gorenstein ring appears to be projective; this probably follows also from Jeremy Rickard's argument.) Second, since this is a local problem, it suffices to prove the case when $R$ is a local, Noetherian, Gorenstein ring of some dimension $d$ (local Noetherian rings have finite Krull dimension). The result is proved by induction on $d$. I will use some results from Chapter 21 of the following. (Aside: This is the second time I have mentioned Chapter 21 on MathOverflow in two days.)

The following addresses your second question in the case that $R$ is commutative, about $\text{Hom}_R(M,N)$. First of all, it is false if there is not some additional hypothesis. Let $R$ be the zero-dimensional, local, Noetherian, Gorenstein ring $k[x]/\langle x^2 \rangle$. Let $M$ and $N$ both be $R/xR = k$. Then $\text{Id}_M\in \text{Hom}_R(M,M)$ is not in the image of the morphism $\text{Hom}_R(M,R)\otimes_R M \to \text{Hom}_R(M,M)$. The only reasonable additional hypothesis I can see is that $N$ has finite injective dimension. However, in this case it seems to me that $N$ is a finitely generated projective $R$-module, so that the problem becomes trivial. I am including the following argument for any case.

<Edit.> The OP confirms that in the commutative case he is considering only regular rings, so then the problem is trivial. I thought the OP was only assuming that $R$ has finite injective dimension (the usual definition of Gorenstein). The OP is interested in the non-commutative case, not the commutative case. At any rate, I am including below the argument in the commutative case when $R$ is not assumed regular, but $N$ is assumed to have finite injective dimension.

The following addresses your second question in the case that $R$ is commutative, about $\text{Hom}_R(M,N)$. First of all, the result is false if there is not some additional hypothesis. Let $R$ be the zero-dimensional, local, Noetherian, Gorenstein ring $k[x]/\langle x^2 \rangle$. Let $M$ and $N$ both be $R/xR = k$. Then $\text{Id}_M\in \text{Hom}_R(M,M)$ is not in the image of the morphism $\text{Hom}_R(M,R)\otimes_R M \to \text{Hom}_R(M,M)$. The only reasonable additional hypothesis I can see is that $N$ has finite injective dimension. However, in this case it seems to me that $N$ is a finitely generated projective $R$-module, so that the problem becomes trivial. I am including the following argument for any case.

Edit. Something is wrong with the base case of the following induction. I will try to fix it.

From the formulation of your question, it sounds like you are interested in associative but not-necessarily-commutative rings. I cannot say anything about those. However, for every commutative Noetherian ring $R$, for all maximal Cohen-Macaulay $R$-modules $M$ and $N$, the natural $R$-module homomorphism, $$\beta_{R,M,N}:\text{Hom}_R(M,R)\otimes_R N \to \text{Hom}_R(M,N),$$ is an isomorphism.

First, since $\beta_{R,M,N}$ is a natural transformation of additive functors, this is obviously an isomorphism when $N$ is a finite free $R$-module. Second, since this is a local problem, it suffices to prove the case when $R$ is a local, Noetherian, Gorenstein ring of some dimension $d$ (local Noetherian rings have finite Krull dimension). The result is proved by induction on $d$. I will use some results from Chapter 21 of the following. (Aside: This is the second time I have mentioned Chapter 21 on MathOverflow in two days.)

In the base case that $d=0$, by Proposition 21.11, both $M$ and $N$ are finite free $R$-modules. Thus by the first observation above, $\beta_{R,M,N}$ is an isomorphism.

Thus, by way of induction, assume that $d>0$, and assume that the result has been proved for smaller values of $d$. Since $R$ is Gorenstein, it is Cohen-Macaulay, so that $\text{depth}_R(M) = d > 0$. So there exists a system of parameters $(x_1,\dots,x_d)$ for $R$. By Proposition 21.9, this system of parameters is simultaneously an $M$-sequence and an $N$-sequence. Denote $\overline{R} = R/x_d R$, $\overline{M} = M/x_d M$, and $\overline{N} = N/x_d N$. In particular, consider the $\overline{R}$-module homomorphism induced from $\beta_{R,M,N}$, $$ \overline{\beta_{R,M,N}} : \text{Hom}_R(M,R)\otimes_R N \otimes_R \overline{R} \to \text{Hom}_R(M,N)\otimes_R \overline{R}.$$ By Proposition 21.13, to prove that $\beta_{R,M,N}$ is an isomorphism, it is equivalent to prove that $\overline{\beta_{R,M,N}}$ is an isomorphism.

Since $R$ is a local Gorenstein ring of dimension $d$ and $x_d\in \mathfrak{m}_R$ is a nonzerodivisor, the quotient ring $\overline{R}$ is a local, Noetherian, Gorenstein ring (there are other references, but this does follow from Exercise 21.20). Also, since $(x_1,\dots,x_d)$ is both an $M$-sequence and an $N$-sequence, by definition, also the induced system of parameters $(\overline{x}_1,\dots,\overline{x}_{d-1})$ is both an $\overline{M}$-sequence and an $\overline{N}$-sequence. Therefore, by Proposition 21.9 again, both $\overline{M}$ and $\overline{N}$ are maximal Cohen-Macaulay $\overline{R}$-modules. Thus, by the induction hypothesis, the $\overline{R}$-module homomorphism,$$ \beta_{\overline{R},\overline{M},\overline{N}}: \text{Hom}_{\overline{R}}(\overline{M},\overline{R})\otimes_{\overline{R}} \overline{N} \to \text{Hom}_{\overline{R}}(\overline{M}, \overline{N}), $$ is an isomorphism. Finally, since $R$, $M$ and $N$ are maximal Cohen-Macaulay $R$-modules, by Proposition 21.12(b) (applied twice), the $\overline{R}$-module homomorphism $\beta_{\overline{R},\overline{M},\overline{N}}$ is equivalent to the $\overline{R}$-module homomorphism $\overline{\beta_{R,M,N}}$. Therefore $\beta_{R,M,N}$ is an isomorphism. By induction, the result holds for all $d$.

Original argument with additional hypothesis -- probably overkill. For every commutative Noetherian ring $R$, for all maximal Cohen-Macaulay $R$-modules $M$ and $N$, assuming that $N$ has finite injective dimension, then the natural $R$-module homomorphism, $$\beta_{R,M,N}:\text{Hom}_R(M,R)\otimes_R N \to \text{Hom}_R(M,N),$$ is an isomorphism.

First, since $\beta_{R,M,N}$ is a natural transformation of additive functors, this is obviously an isomorphism when $N$ is a finite free $R$-module. (Edit. As Jeremy Rickard observes as well, that is probably enough to conclude, since a maximal Cohen-Macaulay module of finite injective dimension appears to be projective.) Second, since this is a local problem, it suffices to prove the case when $R$ is a local, Noetherian, Gorenstein ring of some dimension $d$ (local Noetherian rings have finite Krull dimension). The result is proved by induction on $d$. I will use some results from Chapter 21 of the following. (Aside: This is the second time I have mentioned Chapter 21 on MathOverflow in two days.)

In the base case that $d=0$, by Proposition 21.11, $N$ is a finite free $R$-modules. Thus, by the first observation above, $\beta_{R,M,N}$ is an isomorphism.

Next, by way of induction, assume that $d>0$, and assume that the result has been proved for smaller values of $d$. Since $R$ is Gorenstein, it is Cohen-Macaulay, so that $\text{depth}_R(M) = d > 0$. So there exists a system of parameters $(x_1,\dots,x_d)$ for $R$. By Proposition 21.9, this system of parameters is simultaneously an $M$-sequence and an $N$-sequence. Denote $\overline{R} = R/x_d R$, $\overline{M} = M/x_d M$, and $\overline{N} = N/x_d N$. In particular, consider the $\overline{R}$-module homomorphism induced from $\beta_{R,M,N}$, $$ \overline{\beta_{R,M,N}} : \text{Hom}_R(M,R)\otimes_R N \otimes_R \overline{R} \to \text{Hom}_R(M,N)\otimes_R \overline{R}.$$ By Proposition 21.13, to prove that $\beta_{R,M,N}$ is an isomorphism, it is equivalent to prove that $\overline{\beta_{R,M,N}}$ is an isomorphism.

Since $R$ is a local Gorenstein ring of dimension $d$ and $x_d\in \mathfrak{m}_R$ is a nonzerodivisor, the quotient ring $\overline{R}$ is a local, Noetherian, Gorenstein ring (there are other references, but this does follow from Exercise 21.20). Also, since $(x_1,\dots,x_d)$ is both an $M$-sequence and an $N$-sequence, by definition, also the induced system of parameters $(\overline{x}_1,\dots,\overline{x}_{d-1})$ is both an $\overline{M}$-sequence and an $\overline{N}$-sequence. Therefore, by Proposition 21.9 again, both $\overline{M}$ and $\overline{N}$ are maximal Cohen-Macaulay $\overline{R}$-modules. By Proposition 21.10, since $N$ has finite injective dimension as an $R$-module, also $\overline{N}$ has finite injective dimension as an $\overline{R}$-module. Thus, by the induction hypothesis, the $\overline{R}$-module homomorphism,$$ \beta_{\overline{R},\overline{M},\overline{N}}: \text{Hom}_{\overline{R}}(\overline{M},\overline{R})\otimes_{\overline{R}} \overline{N} \to \text{Hom}_{\overline{R}}(\overline{M}, \overline{N}), $$ is an isomorphism. Finally, since $R$, $M$ and $N$ are maximal Cohen-Macaulay $R$-modules, by Proposition 21.12(b) (applied twice), the $\overline{R}$-module homomorphism $\beta_{\overline{R},\overline{M},\overline{N}}$ is equivalent to the $\overline{R}$-module homomorphism $\overline{\beta_{R,M,N}}$. Therefore $\beta_{R,M,N}$ is an isomorphism. By induction, the result holds for all $d$.