On page 103 of "Theory of Stein Spaces" by Grauert and Remmert is the following theorem:
Let $X$ be a paracompact space and $\mathcal{F}$ a sheaf of abelian groups on $X$. Let $$K_0 \subseteq K_1 \subseteq K_2 \subseteq K_3 \subseteq \cdots$$ a sequence of compact subsets of $X$ such that $K_i \subseteq \text{int} \, K_{i + 1}$ for all $i \in \mathbb{Z}_{\ge 0}$ and $X = \bigcup_{i = 0}^\infty K_i$.
If $n \in \mathbb{Z}_{\ge 2}$ is such that $$ H^n \left( K_i, \mathcal{F} \mid_{K_i} \right) = H^{n - 1} \left( K_i, \mathcal{F} \mid_{K_i} \right) = 0 $$ for all $i \in \mathbb{Z}_{\ge 0}$, then also $H^n \left( X, \mathcal{F} \right) = 0$.
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The proof is by taking a flasque resolution and then a mostly straightforward diagram chase. However, it's not clear to me where we need the fact that $K_i \subseteq \text{int} \, K_{i + 1}$ for each $i \in \mathbb{Z}_{\ge 0}$, or the fact that the $K_0, K_1, K_2, K_3, \cdots$ are compact.
I know that the requirement that each compact set is contained in the interior of the next is necessary: if $X$ is the torus as the quotient $\mathbb{R}^2 / \mathbb{Z}^{\oplus 2}$ and $K_i \subseteq X$ is the image of the square $[0, 1 - 1/i]^2 \subseteq \mathbb{R}^2$ in the quotient, then $\{ K_i \}_{i \in \mathbb{Z}_{\ge 0}}$ is an increasing sequence of compact subsets of $X$ which covers the whole space. Since these are contractible, $$ H^2 \left( K_i, \underline{\mathbb{Z}} \right) = H^1 \left( K_i, \underline{\mathbb{Z}} \right) = 0 $$ for all $i \in \mathbb{Z}_{\ge 0}$.However, it's known that $H^2 \left( X, \underline{\mathbb{Z}} \right) \ne 0$, contrary to the above theorem.
I want to know where the assumption of each compact being contained in the interior of the next is used in the proof. I suspect that there's some technicality about when the restrictions maps of a flasque sheaf are surjective when restricting between non-open set, i.e. maybe if $K_1, K_2 \subseteq X$ compact and $K_1 \subseteq K_2$, then for a flasque sheaf $\mathcal{I}$, the restriction $\mathcal{I}(K_2) \longrightarrow \mathcal{I}(K_1)$ is not necessarily surjective if $K_1 \not\subseteq \text{int} \, K_2$. If this is so, I'd appreciate a counterexample because it seems to me that restricting between compact subsets in a flasque is always surjective.
The proof in the book uses the following argument:
If $\beta_{i} \in \mathcal{F}(K_{i})$ such that $\beta_{i} = r_{i}(\beta_{i+1})$ where $r_{i}$ is induced by inclusion $K_{i}\subset K_{i+1}$ then there exists section $\beta\in \mathcal{F}(X)$ such that $\beta$ restricts to $\beta_{i}$ in $K_{i}$.
If condition $K_{i} \subset intK_{i+1}$ is omitted then it is not true. For instance, consider the example you gave with torus $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ and $K_{i} = [0,1-1/i]^{2}$. Take sheaf $\mathcal{F}$ of continous functions. Then restriction of function $f(x)=x$ on each $K_{i}$ defines sections of $\mathcal{F}(K_{i})$ which satisfy condition above. But this sections can not be glued to a section of $\mathcal{F}(\mathbb{T}^{2})$.
The main reason why this does not happen when $K_{i} \subset intK_{i+1}$ is the fact that you can find system of open sets $U_{i} \subset U_{i+1}\subset\dots$ which meet the conditions
So there is no additional intersections between $U_{i}$ and $U_{j}$ and that is why you can glue sections $\beta_{i}$.