Is every injective ring homomorphism an automorphism?

No, consider $R=\Bbb Z[X]$. Let $\phi$ be defined by $X\mapsto X^2$, i.e. $$\phi\left(\sum_{i=0}^n a_iX^i \right)=\sum_{i=0}^n a_iX^{2i}$$ Then $\phi$ is injective but its image does not contain $X$.
It is also not true if we replace 'injective' with 'surjective'. To see this, let $R=\Bbb Z[X_1,X_2,X_3,\dots]$ where $\phi$ is given by $X_1\mapsto 0$ and $X_i\mapsto X_{i-1}$ for $i>1$.
So rings can be isomorphic to both proper subrings and proper quotients.