I'm not entirely certain that this was the first use of the phrase, but the "War on Drugs" started before I was even born (the seventies). To say it is "beginning" to get annoying is to say that my avocado colored bathroom is beginning to look a little dated.
The War on Poverty is the name for legislation first introduced by United States President Lyndon B. Johnson during his State of the Union address on January 8, 1964. http://en.wikipedia.org/wiki/War_on_Poverty