Let $X$ be a topological space and $\mathcal{F}$ an abelian sheaf on $X$. Furthermore let $0=X_0 \subset X_1 \subset X_2 \subset \ldots$ be an increasing sequence of subspaces of $X$ such that $X=\bigcup X_i$ $(= \varinjlim X_i)$. By functoriality of sheaf cohomology we have for each $q \in \mathbb{N}$ natural maps $H^q(X_{i+1},\mathcal{F}_{|X_{i+1}})\rightarrow H^q(X_i,\mathcal{F}_{|X_i})$ and especially we can construct the inverse limit $\underset{i}{\varprojlim} H^q(X_i,\mathcal{F}_{|X_i})$ along these maps. By the universal property of the inverse limit and again by the functoriality of sheaf cohomology we have natural maps $\phi_q:H^q(X,\mathcal{F})\rightarrow \underset{i}{\varprojlim} H^q(X_i,\mathcal{F}_{|X_i})$.
Is $\phi_q$ in general an isomorphism? Or if not, under which (mild) assumptions this is known to be an isomorphism?
First of all, be careful about how you name your things : the cohomological index $i$ is probably not the same as the indexing $i$.
This map will generally not be an isomorphism : indeed $\Gamma(X, \mathcal F) = \varprojlim_i \Gamma(X_i, \mathcal F_{\mid X_i})$ so as functors on $Sh(X,\mathbf{Ab})$, $\Gamma(X,-) = \varprojlim_i \Gamma(X_i, -)\circ\mathrm{res}_{X_i}$
Sheaf cohomology is the right derived functor of the RHS, so it's the right derived functor of the LHS.
You will nevertheless have (under mild hypotheses) something like $R\Gamma(X,-) = R\varprojlim_i R\Gamma(X_i,-)\circ \mathrm{res}_{X_i}$ (because $\mathrm{res}_{X_i}$ is exact: it's of the form $f^{-1}$ for $f:X_i\to X$ the inclusion)
If you don't like derived functors, this translates into a spectral sequence with $E_2^{p,q} = R^p\varprojlim_i H^q(X_i,\mathcal F_{\mid X_i})\implies H^{p+q}(X,\mathcal F)$
Since we're dealing with abelian groups, $R^p\varprojlim_i$ vanishes for $p>1$ so the spectral sequence is particularly nice (it has only two columns), and in fact for degree reasons all its differentials are $0$, so you have a $E_\infty^{p,q} = E_2^{p,q}$.
This gives short exact sequences $0\to \varprojlim^1_i H^q(X_i, \mathcal F_{\mid X_i})\to H^{q+1}(X,\mathcal F)\to \varprojlim_i H^{q+1}(X_i,\mathcal F_{\mid X_i})\to 0$
where $\varprojlim_i^1$ is the first derived functor of $\varprojlim_i$.
If your system of subspaces is particularly nice (nice enough to have the spectral sequence above; and for the system $(H^q(X_i,\mathcal F_{\mid X_i}))_i$ to be $\varprojlim_i$-acyclic (e.g. if it satisfies the Mittag-Leffler condition; or if it's particularly nice and the transition maps are surjective)), then you will indeed have an isomorphism, but in general there's no reason to expect it.