I would like to know if there are submersions $\pi : F\hookrightarrow M\to B$ (with fiber $F$) from a closed manifold $M$ to a manifold $B$ with boundary.
I really suspect that the answer is no, since $M$ is compact it is possible to trivialize $\pi$ (in the sense of fiber bundles trivializations). So it would imply that $B$ can't have boundary.