Let $\mathcal O$ be a principal ideal domain with field of fractions $F$. Let $A$ be an algebra over $F$, and $A_0$ an $\mathcal O$-subalgebra of $A$. Since $A_0$ is torsion free over $\mathcal O$, it must be a flat.
Is the natural $F$-algebra homomorphism $A_0 \otimes_{\mathcal O} F \rightarrow A$ always injective?
Can we say that $A_0 \otimes_{\mathcal O} A_0 \rightarrow A \otimes_F A$ is injective?
I am rusty at making these kinds of arguments, but I recall that with flat algebras you can generally declare a lot of things to be injections.
Yes: the homomorphism $A_0\otimes F\to A$ is just obtained by tensoring the inclusion map $A_0\to A$ with $F$ (since $A\otimes F\cong A$), and so it is injective because $F$ is flat over $\mathcal{O}$. Similarly, the map $A_0 \otimes A_0 \rightarrow A \otimes A$ can be written as the composition $A_0\otimes A_0\to A_0\otimes A\to A\otimes A$ where the first map is the inclusion $A_0\to A$ tensored with $A_0$ and the second map is the inclusion $A_0\to A$ tensored with $A$, both of which are injective since $A_0$ and $A$ are flat. (Note that $A\otimes_F A$ and $A\otimes_\mathcal{O} A$ are the same thing so you may as well tensor everything over $\mathcal{O}$.)