Is one obligated to wear Tefillin when saying Shema in the morning before Davening?
No, and so is the custom of most Chassidim. However some Chassidim are accustomed to don Tefillin and recite Shema before eating prior to Davening, and so the was the directive of the Rebbe Rashab and Rebbe Rayatz. This is not the widespread Chabad custom.
 58/4; However see 66/11; Torah Or Vayakhel
The reason: It is not considered false testimony to say the morning Shema prior to prayer, without Tefillin, as one will wear Tefillin later on. [Rebbe in Toras Menachem 28th Sivan 1952]
 Os Chaim 25/6; Rebbe in Toras Menachem 28th Sivan 1952
 Directive of Rebbe Rashab in Kuntrus Eitz Chaim 25; Igros Kodesh of Rebbe Rayatz letter 3302