Concatenation of the intersection of two languages
First, you may notice that there is a one-sided containment here. It always holds that 
$(L_1\cap L_2)L_3\subseteq L_1L_3\cap L_2L_3$ (prove it!)

From this, you see that in order for the equality to fail, you need to get a strict containment.
Intuitively, this means that you need the concatenation with $L_3$ to add words to both $L_1$ and $L_2$.

One way to achieve this is, for example, to first let $L_1\cap L_2=\emptyset$, and then "tailor" $L_3$ to fix this emptiness. 

A concrete example is below (hidden as spoiler, hover mouse to see it):


  $L_1=\{a\}$, $L_2=\{aa\}$, $L_3=\{a,\epsilon\}$.