I read Igros Moshe forbids listening to music with idolatrous intent. What are some sources, if any, that allow listening to, or playing/performing Christian classical music, such as Bach's or Handel's oratorios? And what about Christian music without words, or words exclusively from Tanach?
I couldn't find a specific source, but the following may be relevant.
One of my Rabbeim once explained to me regarding classical music that "music is not mekabel tumah". However, if there are identifiable religious connotations then it may be problematic. For example, if you listen to a famous religious choral work (Bach's Mass B minor or Mozart Requiem, etc.) but replaced all the words with "lah lah lah", then although the narrative of the words is gone, the essential nature of the piece may still remain when you listen to it. I would imagine that the images and feelings that the piece evokes changes from person to person and would make a difference as to whether it may be permissible or not.
Having said this, there are several classical (small 'c') music forms which almost always comprise a religious (Christian text) element, e.g., Requiem, Mass, Chorale, Hymn, Motet, Gregorian Chant, etc.
There are two sides: If you are a musical expert and can tell the precise form of the music from the structure, tone, etc. then you may be able to directly identify if you are listening to a religious work. You may even come to associate a particular melody with the precise words as they correspond to the Christian bible, for example. On the other hand, if you are ignorant of classical music entirely, there may be no Christian connotation even to works that have an explicit religious nature. The same music will affect different people differently.
A final decision regarding the matter is best left to the individual person, their situation and probably in discussion with a Rav that they are acquainted with.