The following statement should be true:
Let $N^n\hookrightarrow M^m$ be a smooth submanifold with codimension $k:=m-n$. Then $H^\bullet(M\setminus N\hookrightarrow M)$ is injective for $\bullet\le k-1$ and surjective for $\bullet\le k-2$.
The reason is mainly the Thom isomorphism $H^{\bullet-k}(N)\cong H^\bullet(M,M\setminus N)$, which gives rise to exact sequences
$$\dotsb\to H^{\bullet-k}(N)\to H^\bullet(M)\to H^\bullet(M\setminus N)\to H^{\bullet+1-k}(N)\to\dotsb$$
and now $H^{\bullet-k}(N)=0$ for $\bullet\le k-1$. Unfortunately, this argument uses the existence of tubular neighborhoods. Is there any generalisation for subcomplexes, e.g. if $N$ is a simplicial subcomplex?