If you are using ordinary algebras, and you are not assuming that your homomorphisms have to preserve the identity of the ring, then there is always the map $a\mapsto (a,0)$ that injects $A$ into $A\times B$. However, I'm guessing you want your maps to preserve identity, since otherwise zero homomorphisms would be everywhere.
If you want your map to preserve identity, then finding an example of $A$ that does not embed into $A\times B$ becomes really simple. Pick $A$ and $B$ to have two different coprime finite characteristics, say $7\cdot 1_A=0_A$ and $10\cdot 1_B=0_B$. Then the characteristic of the product ring is 70. But then if $\phi$ was an additive map preserving identity, $\phi(0)=\phi(7\cdot 1_A)=7\phi(1_A)=7\cdot 1_{A\times B}\neq 0$, a contradiction.