Let $(\mathcal{C},\otimes, I)$ be a symmetric monoidal closed category (with $\otimes$-unit $I$).
Suppose that $\mathcal{D}$ is a (full) subcategory of $\mathcal{C}$, which contains $I$, and which is closed under $\otimes$, i.e. $A,B\in\mathcal{D}$ implies $A\otimes B\in \mathcal{D}$.
Does it follow that $(\mathcal{D},\otimes)$ becomes a symmetric monoidal closed category itself?
No, $\mathcal{D}$ will usually not be closed. For instance, let $\mathcal{C}=\mathtt{Set}$ and $\otimes=\times$, and let $\mathcal{D}$ be the full subcategory of countable sets. Then $\mathcal{D}$ is a monoidal subcategory, since a product of countable sets is countable. But an exponential of countable sets need not be countable, so there is no reason to expect $\mathcal{D}$ to be closed. To explicitly prove that it isn't, note that if $X$ were an internal Hom-object from $\mathbb{N}\to\mathbb{N}$, then there would need to be a map from $1$ to $X$ for each map from $1\times\mathbb{N}\to\mathbb{N}$, so $X$ would need to have uncountably many points.