This characterizes normality. That it implies normality was observed above by Remy.
Conversely, assume $f$ vanishes on $A\cap B$. Define $h:A\cup B\to\mathbb{R}$ by $h(x)=f(x)$ if $x\in A$ and $h(x)=0$ if $x\in B$. By the Tieze-Urysohn theorem $h$ has a continuous extension $H:X\to\mathbb{R}$. That extension belongs to $J_B$. But then $G=f-H$ belongs to $J_A$ because $f(x)=H(x)$ on $A$. We have $f=G+H$.