Let $C$ be a site, i.e. a category equipped with a Grothendieck topology (or coverage).
Let $A,B$ be complete categories, and let $T : A \rightarrow B$ be a continuous functor.
My question is, if $F$ is an $A$-valued sheaf on $C$, is $T \circ F$ necessarily a $B$-valued sheaf on $C$?
Note: if $C$ is e.g. the category of open sets of a topological space $X$, with the usual Grothendieck topology (I think Prop 2.11 here gives a more general case), then the sheaf condition just enforces preservation of certain limits (of $C^{\mathrm{op}}$; see Prop 2.10 at the previous link), hence the sheaf condition should also hold for $TF$ as $T$ is continuous.
I was wondering if the above also holds for sheaves on general sites $C$? If yes, how would we show this explicitly, in the general abstract case?
This is correct for the reason that you describe, as long as it is understood what is meant by a sheaf with values in a general complete category $A$.
I will assume the site has pullbacks. However the ncatlab (Def. 2.6 here) uses the Yoneda embedding to write the sheaf condition as a limit when the site does not have pullbacks---at the cost of assuming more structure on the category of values $A$.
Given a site with pullbacks (a category $C$ with pullbacks equipped with a Grothendieck topology $J$), a contravariant functor $P: C^\text{op} \to A$ is a sheaf if, for every object $X \in C$ and every covering family $f_\alpha: U_\alpha \to X$, the following diagram is a limit in $A$: $$ P(X) \xrightarrow{\rho} \prod_{\alpha} P(U_\alpha) \rightrightarrows \prod_{\alpha,\beta} P(U_\alpha \times_X U_\beta), $$ where $\rho$ is the morphism induced by the restrictions $P(f_\alpha): P(X) \to P(U_\alpha)$. The other two morphisms on the diagram come from applying $P$ to the projections $U_\alpha \times_X U_\beta \to U_\alpha$ and $U_\alpha \times_X U_\beta \to U_\beta$.
Since $T: A \to B$ preserves limits, we obtain a limit diagram in $B$: $$ TP(X) \xrightarrow{T\rho} \prod_{\alpha} TP(U_\alpha) \rightrightarrows \prod_{\alpha,\beta} TP(U_\alpha \times_X U_\beta), $$ for any object $X$ and any covering family of $X$. Therefore, $TP: C^\text{op} \to B$ is a sheaf.
An example of this in action is restriction of scalars (as long as you believe that certain completeness conditions are satisfied): given a morphism of rings $f: R \to S$ we obtain a functor $T: S\text{-Mod} \to R\text{-Mod}$. $T$ preserves limits (it has a left adjoint, called extension of scalars), so it sends sheaves to sheaves, e.g., a sheaf of $R$ modules is also a sheaf of $\mathbb{Z}$-modules (abelian groups).