Does the Christian Bible have Kedushah? It does have the entire Tanach in it, just with an additional part. Does this make it lose its Kedushah? I'm asking about a single-volume printing.
יג ספר תורה תפילין ומזוזות שכתבן מין, יישרפו. כתבן גוי, או ישראל משומד, או מוסר, או עבד, או אישה, או קטן--הרי אלו פסולין וייגנזו: שנאמר "וקשרתם . . . וכתבתם" (דברים ו,ח-ט; דברים יא,יח-כ)--כל שמוזהר על הקשירה ומאמין בה, הוא שכותב. נמצאו ביד מין ואין ידוע מי כתבן, ייגנזו; נמצאו ביד גוי, כשרים. ואין לוקחין ספרים תפילין ומזוזות מן הגויים ביתר על דמיהן, שלא להרגיל אותם לגונבם ולגוזלם.
A Torah scroll, Tefillin, or Mezuzah written by a heretic -- burn it!... If you find one in the possession of a heretic and don't know who wrote it, bury it; if in the possession of a non-Jew, assume it's kosher.
If we destroy a heretic-written sefer torah, then it seems that that particular sefer torah didn't have kedusha even though sifrei torah usually do. Similarly, even though a Tanakh has kedusha in general, I would argue (from the sefer torah) that it doesn't when written by a heretic (min, in the Rambam passage). So we must ask: is a Christian bible the product of a heretic?
There is some ambiguity about the definition of heresy:
In talmudic literature a number of terms are used to refer to heretics, min, apikoros, kofer, and mumar, each of which also has other meanings. Min is the most common term and the one that appeared originally in the 12th petition of the daily *Amidah. Some identify the talmudic minim with the Judeo-Christians, others with unspecified groups who denied rabbinic authority and/or the belief in the coming of the Messiah. There is an early tradition that there were 24 groups of minim as early as the destruction of the Second Temple (TJ, Sanh. 10:29c). Among the errors of the minim, the Talmud lists denial of God's unity; belief in an independent divinity of evil; the portrayal of God as a cruel jester (Sanh. 38b–39a); and the denial of Israel's chosenness (Sanh. 99a), physical resurrection, and the coming of the Messiah (Sanh. 91a). *Maimonides identified minut with atheism, with the denial of God's unity and incorporeality, with the denial of creation ex nihilo, and with the belief in a power intermediary between God and man (Yad, Teshuvah 3:7). (source)
According to the talmud minim include those who deny God's unity or believe in an independent, divine evil being. Both of those are core tenets of Christianity. What is not clear, though, is whether the talmud refers only to Jews; can a non-Jew be a min at all?
According to this ruling attributed to R' Riskin, a missionary bible should be burned (all parts), and a bible written on a heretical basis is not sacred. So bibles that are produced or distributed withe purpose of evangelizing Jews can (and according to this should) be destroyed, including the Tanakh passages.
A further complication: the text in a Christian translation of the Tanakh has been altered. This is widely known but the practice continues. I would think that this would also affect any kedusha it might have had, though I can't bring a source.
Putting all this together:
a Christian Tanakh text written by apostate Jews would certainly not have kedusha (it's like the sefer torah)
a Christian Tanakh text written by gentiles with an intent to evangelize has no kedusha according to R' Riskin
any other Christian Tanakh text probably doesn't have kedusha, unless they've actually reprinted the original Hebrew and not a translation
For a practical ruling you should consult your rabbi.