by substituting $A$ with $A-B$ and $B$ by $-B$ your condition will be equivalent to the following: $||P_{A+B}x||\leq ||P_Ax||+||P_Bx||.$ the last one is not true in general (for non-selfadjoint matrices), for example it does not hold for $A=((0,1),(0,0))$, $B=((1,1),(1,1))$ and $x=\frac{1}{\sqrt{2}}(1,-1)$.

by substituting $A$ with $A-B$ and $B$ by $-B$ your condition will be equivalent to the following: $||P_{A+B}x||\leq ||P_Ax||+||P_Bx||.$ the last one is not true in general (for non-positive matrices), for example it does not hold for $A=((0,1),(0,0))$, $B=((1,1),(1,1))$ and $x=\frac{1}{\sqrt{2}}(1,-1)$.