My view is I have to be able to feel the song rather than believe it – it's response at an emotional level that matters. Some songs I can feel even if I don't believe in them. I do, however, also try to take into consideration what the audience may take from my singing of the song. If I disagree with the topic covered by the lyrics, and the club/singaround is the sort where I can introduce my song, I can put it in context; if I'm just supposed to get on with it and sing, I may apply a different filter. I don't hunt, for example, and don't approve of "sport" hunting but I'll sing hunting songs at some places because I know they won't offend (and they do have great choruses for joining in with.) At other places I'd only sing them if I could explain my disagreement with the practice but appreciation of the history, pageantry and "country" culture that spawned them.
One example of a "hunting" song I sing is a version of "The Coast of Peru". I'll only sing it if I can put it into context as I can find no justification for whaling these days but there was a time when it was integral to the survival of some communities. I actually treat the song ("feel" it) as an anti-hunting song that sits at the time when whaling was changing from a means of food-gathering and subsistence to a time of greed and extermination. With a suitable introduction, lines like "Which made her to vomit and the blood for to spout" and "And you that wants money I'd have you to go; To the coast of Peru me boys, where the whalefish they blow" carry a real message for me. I don't "believe" the song but I can feel both the hard life of the whalers and the greed for money that turned whaling into an industry that's set to exterminate species.
P.S. And where would my repertoire be if I ditched all the hunting songs, incest songs, infidelity songs, murder ballads, slaving (Flying Cloud) songs, Man Selling his Wife songs, songs extolling delicacies like Lumpy Tums, etc?
P.P.S. I share McGrath of Harlow's view about DITROT. Or should I also be offended by its condemnation of students! (On second thoughts….)