The Aruch HaShulchan writes (Y.D. 265:38):
וכתבו במנהגים שמלין קודם "עלינו", ואין חולצין התפילין עד אחר המילה (ש"ך סעיף קטן כ"ד). וגם אומרים פיוטים (שם). ואצלינו לא נהגו בפיוטים, וכן למול קודם "עלינו" אין נוהגים, וכן בחליצת התפילין, משום דעל פי רוב אין מלין בבית הכנסת רק בבית המילה. ולכן גם כשמלים לפרקים בבית הכנסת אין המנהג כן.
That the custom mentioned in the Shach (brought in other answers) to leave on the Tefillin is no longer operative because the Bris is no longer regularly done in Shul. Thus even when, on occasion, it is done in Shul the Tefillin are not left on.
This does not explain announcing that one should specifically take them off, though.
The Kaf HaChayim (25:92) actually says that the Sandek, Mohel and Father specifically should put Teffilin on for the Bris (by implication even if the Bris is not being done right after a Shacharis minyan, where everyone else - it is obvious, he says - would not have them on). Right after a Shacharis minyan, he expects everyone to keep them on for the Bris. But they should not be put back on on Rosh Chodesh for the Bris (25:96) (until Mincha) for Kabbalistic reasons.
Kitzur S.A. Yalkut Yosef (O.C. 28:10) says that if the Sandek or Father leave their Teffilin on they have what to rely on (although some object to this), but the accepted (Sefardic, I guess) practice as seen by many great Rabbis is to not wear Teffilin during the Bris. In any event he says that on Rosh Chodesh Tefillin should not be put back on for the Bris.