$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras. Is it true that $A^+  \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$. 
I read in Murphy 2.1.11 that this is true if $B$ is unital and $A$ contains the unit of $B$. 
Does it make sense to look at unitizations of $A$ and $B$ if they are non-unital ?

Yes, that makes sense. Murphy does this at the beginning of section 2.2, on pages 44 and 45. In the non-unital case we get $\sigma_B(b) \cup \{0\} = \sigma_A(b) \cup \{0\}$, but this does not mess up positivity.
In order to understand the answer Martin Argerami gives, see Murphy theorem 2.2.5(1).
