Let $M$ be a compact submanifold (with boundary) of $S^n$ realised as the intersection of some other compact manifolds $M_i\subset S^n$ so $M=\cap M_i$ and $M_i\subset Int(M_{i-1})$. Then we have:
$\pi_1(S^n\setminus M)= \varinjlim \pi_1(S^n\setminus M_i)$
Where the direct limit is taken with respect to the inclusions $S^n\setminus M_i \subset S^n\setminus M_{i+1}$.
Then, what are some conditions on $\pi_1(S^n\setminus M_i)$ for $\pi_1(S^n\setminus M)$ to be nontrivial?
For example is it enough to find some non-zero element $g\in \pi_1(S^n\setminus M_1)$ since it will remain non-zero under inclusion? Or is there an example of such a sequence where each $\pi_1(S^n\setminus M_i)$ is non-zero but the direct limit is?
For an arbitrary sequence of groups this definitely can happen:
$$ G\rightarrow G\rightarrow G\rightarrow \ldots$$ $$\mathbb{Z}_4\rightarrow \mathbb{Z}_4\rightarrow \mathbb{Z}_4\rightarrow \ldots$$
Where in the first chain the maps are zero and in the second chain the maps are multiplication by $2$. Then in both cases the direct limit is $0$.
It is possible for $\pi_1(S^n\setminus M)$ to be trivial even if $\pi_1(S^n\setminus M_i)$ is nontrivial for all $i$. For instance, let $n=3$ and let each $M_i$ be a solid torus, with $M_{i+1}$ being contained in a ball that is contained in $M_i$, with the radii of the balls going to $0$. Then $\pi_1(S^3\setminus M_i)\cong\mathbb{Z}$ for each $i$, but $M=\bigcap M_i$ is just a single point so $\pi_1(S^3\setminus M)$ is trivial. In this case the maps $\pi_1(S^3\setminus M_i)\to \pi_1(S^3\setminus M_{i+1})$ are the zero map $\mathbb{Z}\to\mathbb{Z}$ for each $i$.
In general, just from the definition of the direct limit, you can see that $\pi_1(S^n\setminus M)$ will be nontrivial iff there is an element $g\in \pi_1(S^n\setminus M_i)$ for some $i$ whose image in $\pi_1(S^n\setminus M_j)$ is nontrivial for all $j\geq i$. I doubt there is much more you can say than this in general, though if you know something much more specific about the $M_i$ there might be some other useful criterion you could come up with. In general, just knowing the groups $\pi_1(S^n\setminus M_i)$ abstractly will not tell you anything about the nontriviality of $\pi_1(S^n\setminus M)$ (since $S^3\setminus M_i$ could always be contained in a contractible subset of $S^3\setminus M_{i+1}$); you need to know something about the maps $\pi_1(S^n\setminus M_i)\to \pi_1(S^n\setminus M_{i+1})$.