Let $X$ be a topological space with an open covering $\mathcal{U}=\{U_i\}_{i\in I}$. Let $\mathcal{V}$ be a subcovering of $\mathcal{U}$, i.e. there exists a subset $J\subset I$ s.t. $\mathcal{V}=\{U_i\}_{i\in J}$. Then $\mathcal{V}$ refines $\mathcal{U}$ and for every sheaf $\mathcal{F}$ on $X$ and all $p\geq 0$, there exists a canonical map of Cech cohomolog groups $$H^p(\mathcal{U},\mathcal{F})\to H^p(\mathcal{V},\mathcal{F}).$$
I believe this map is an isomorphism but can't think of a proof or find a reference.
The context of this question comes from this problem.
This is not true in general. For instance, let $X=S^1\subseteq\mathbb{C}$ be the circle, define $U_0=S^1\setminus\{1\}$ and $U_1=S^1\setminus\{-1\}$, and let $\mathcal{U}=\{U_0, U_1, S^1\}$ and $\mathcal{V}=\{U_0, U_1\}$. Then $H^1(\mathcal{V},\mathbb{R})\cong \mathbb{R}$, while $H^1(\mathcal{U},\mathbb{R})\cong 0$ (these are mental computations, so you may want to check this).