Look at the meaning and use of indiferente in the DLE:
indiferente 3. adj. Que no despierta interés o afecto. Ese hombre me es indiferente.
As you see, in present "ser indeferente" (not to arouse interest or affection) is commonly used with the structure me/te/le/les/nos... + es + indiferente.
Ese hombre me es indiferente -> I am indifferent to that man
La guerra me es indiferente -> I am indifferent to the war
So with "Que la guerra no me sea indiferente", the author expresses that he does not want to be indifferent to the war.