I think the best available statement is as follows. Suppose that $E$ and $F$ have the property that whenever $F\wedge X=0$ we also have $F\wedge L_EX=0$. (This holds if $L_E$ is smashing, for example when $E$ is the Johnson-Wilson spectrum $E(n)$.) Then there is a natural homotopy pullback square
$$ \begin{array}{ccc}
L_{E\vee F}X & \rightarrow & L_EX \\
\downarrow & & \downarrow \\
L_FX & \rightarrow & L_EL_FX
\end{array}
$$
Note that $L_{E\wedge F}X$ does not occur here. Probably the most important example is where $E=E(n-1)$ and $F=K(n)$ so $E\vee F$ is Bousfield equivalent to $E(n)$ but $E\wedge F=0$ and also $L_FL_E=0$ (but $L_EL_F\neq 0$).
For another important example, we can take $E=S\mathbb{Q}$ and $F=S/p$ so $E\vee F$ is Bousfield equivalent to $S_{(p)}$. In this case $L_{E\vee F}X=X_{(p)}$ and $L_EX=X\mathbb{Q}$ and $L_FX=X^\wedge_p$ and $L_EL_FX=(X^\wedge_p)\mathbb{Q}$. This gives the $p$-local arithmetic fracture square. For the global arithmetic fracture square, take $F=S(\mathbb{Q}/\mathbb{Z})$ (which is Bousfield equivalent to $\bigvee_pS/p$) instead.
I think that these ideas are all due to Mike Hopkins, but I don't remember what is the best place to read about them. I think there is a good paper by Mark Hovey.
