How are diffeomorphisms related to tangent spaces?
If $F: M \rightarrow N$ is diffeomorphism, then what does this say about $dFp : T_pM \rightarrow T_{F(p)}N$?
What I intuitively think is that if $F$ is diffeomorphic, i.e. differentiable, then this could "transfer" properties that are retained in derivatives to $dFp$.
However, I'm unsure as to what these properties are and why?
Particularly, why does e.g. $F$ invertible $\implies$ $dFp$ invertible?
You should be able to find this fact in any book on smooth manifolds. It is just chain rule: $d(G\circ F)_p=dG_{F(p)}\circ dF_p$. Combining with that $d(id_M)_p=id_{T_pM}$ for the identity map $id_M$ on $M$, this makes the derivative $F\mapsto dF_p$ a covariant functor, and covariant functor in general sends isomorphisms to isomorphisms, by the following observation:
If $G$ is the inverse of $F$, then $id=d(G\circ F)=dG\circ dF$, while $id=d(F\circ G)=dF\circ dG$, i.e. $dF$ and $dG$ are inverse functions of each other.