Why make the phase of an FIR filter linear?
Because linear phase of a filter results in constant group delay according to the relation between group delay $\tau$ and phase response $\phi(\omega)$ of the filter: 
$$ \tau = - \frac{ d\phi(\omega) }{ d\omega} $$

But then you will ask: why then constant group delay? And the short answer will be that constant group delay will preserve the relative positions (in time or space) of different frequency (narrow band) components of the signal to each other. 

Many applications, such as image processing, may require such a fidelity on the processed signals. But not all, and to some degree, non-constant group delay such as provided by IIR filters are well tolerated in applications like audio...