Yes you need to feel a song when you sing it, but its a lot easier to feel the song if you also believe in the sentiments.   I sing a variety of songs, but when I sing anti-war songs or trade union songs I sing them with a lot more passion than I do others.   

Indeed when I've sang the Blackleg Miner I've been told that I sing the last verse with a lot of feeling - Thats because I believe in it.

Paulo
p.s. I've never sang "Will the circle be unbroken" either, but I sing "Miner's Lifeguard" which I do with passion.