Usually I didn't send those things to professors until I was no longer in school. It just felt odd and questionable and I preferred to keep things strictly professional. Since I had professors of various faiths or non-faiths, I would send a beautiful seasonal card of something like a snowy orchard of red apples with icy branches for example, nothing with the connotation of a particular religion. Even a Santa picture could be a little skewed, but this is all entirely up to you. If you did send such a card, make sure the text (printed or your own words) say anything about Christmas. It's just about being polite, ultimately.
Two other reasons not to: you might give the impression of being heedless and not so intelligent (or having a brain full of doggy fluff), as some professors are that judgmental or snobby; second: you will be perceived as brown-nosing, trying to earn future good grades, favoritism, or over-the-top professional or grad school recommendations from them.