Let $J$ be a small discrete category, $B$ be a category with all products, and $F: J \rightarrow B$ by $i \mapsto b_i$. Let $\prod b_i$ be the limit of $F$ with limiting cone $v : \prod b_i \dot \rightarrow F$. If $H: B \rightarrow C$ is another functor, what must be true of $H$ so that
$$Hv : H(\prod b_i) \dot \rightarrow HF$$ is a limiting cone where $$H(\prod b_i) = \prod H(b_i) \text{ or } H(\prod b_i) \cong \prod H(b_i)$$ ?
First of all, the statement $H(\prod b_i) = \prod H(b_i)$ just doesn't make sense. Products are only defined up to isomorphism, so it is only meaningful to ask whether they are isomorphic to things, not whether they are equal to things. In some contexts you might fix one particular product to call "the product", so that it is meaningful to ask whether equality holds, but it will basically never hold. If you ever see anyone writing $=$ in this context, that is probably just lazy notation for "isomorphic" (or maybe "naturally isomorphic").
The statement $H(\prod b_i) \cong \prod H(b_i)$ makes sense, but still isn't a very natural thing to talk about. You pretty much never care about whether $H(\prod b_i)$ is isomorphic to $\prod H(b_i)$ by any map at all. Rather, you care about whether it is isomorphic in a way that is "compatible" with the projections from the products.
What this compatibility means is exactly that $Hv$ is a limiting cone. If $Hv$ is a limiting cone, then that means $H(\prod b_i)$ is a product of the objects $Hb_i$, and so it is isomorphic to any chosen product of them and we can say $H(\prod b_i) \cong \prod H(b_i)$.
So, then, when will $Hv$ be a limiting cone? There isn't any particularly good answer to that. It's a limiting cone when it's a limiting cone. Your question is kind of like asking: if $f:\mathbb{R}\to\mathbb{R}$ is a function and $x,y\in\mathbb{R}$, when is $f(xy)=f(x)f(y)$? It's true when it's true, and if it happens to be true that is a useful fact, but there isn't any simpler way to describe when it's true. It's just a question you can ask and answer of each individual case you encounter.
One important sufficient condition for it to be true is if $H$ has a left adjoint. In fact, a functor which is a right adjoint (i.e., which has a left adjoint) always preserves all limits (in this sense of sending limiting cones to limiting cones), not just products.