Rabbi Moshe Feinstein in Igrot Moshe Yore Deah vol 2 no 111 states:
- Music (with or without words) performed to honor a religious diety is
- Music with words of religous praise are prohibited even when performed
in a secular setting. No distinction is made regarding language or
- Religious music without words of praise in a secular setting (aside from any
problems associated with music in general) is permitted but R. Feinstein
calls it a "davar mechuar" - an ugly/disgusting thing. The instruments
used cannot be instruments generally used for religious purposes.
In addition, in responsa #56, R' Feinstein prohibits listening to Christian religious music.
(Source: http://ottmall.com/mj_ht_arch/v9/mj_v9i98.html#CWQ )
The Mishnah Berurah (53 s.k. 82), based on the Bach (Shu"t Bach haYeshanim 127), says it is permissible to listen to the song unless it was composed for\is primarily sung in Christian religious services.
Please see the responsa inside for details, and ask your LOR for an actual p'sak.
With regards to the possible influence this music could have if you're exposed to it, a writer on Chabad.org states that
On a more Kabbalistic plane, the fifth Lubavitcher Rebbe explains that a composer of music invests his or her very self into the work. The music is an expression of the composer's soul, and listening to music connects the listener's soul to that of the composer.
Another website quotes the Mishna Berura (560 note 25 in Shaar HaTziyun, Laws of Tisha b'Av), who says in the name of the Shelah that non-Jewish tunes can have a negative influence on one's neshama, even if they don't understand the words. (This idea is also mentioned on AskMoses.com and on Ohr Sameach's website.)
The Chabad article continues
In light of this, do you really want to give yourself a soul-connection to just anyone? Especially if the composer is an individual whose spirituality and values are suspect at best?
(However, this sentiment should be tempered by the fact that almost all of the popular Christmas music was composed by Jews. See these articles for details.)
Halachipedia - Listening to Music (especially the "links" section) for the Halachot of listening to music in general.
Rambam's Commentary on the Mishna, Avot 1:17. Loosely translated and explicated here: http://www.torah.org/learning/pirkei-avos/chapter1-17.html
[The Halakhic portion of this answer is cross posted here.]