Embedding a graph as a collection of interior-disjoint tetrahedra
In Hardness of embedding simplicial complexes in $\mathbb{R}^d$ it is stated that $\mbox{EMBED}_{3\rightarrow3}$ is at least as hard as recognizing a 3-sphere, which is known to be in NP, but not known to be in P. They continue to say that for all we know, the problem may be undecidable.

EDIT: Update. Actually, my answer applies to PL embeddings. For linear embeddings the problem is known to be in PSPACE. I don't know if anything else is known.