Question: Can I use my old tefillin bag to keep my shofar in?
Answer: The Gemara (Sanhedrin 48a) teaches that a cloth that was designated for and then used to wrap one’s tefillin, may not later be used to wrap money. Once the cloth has been used for a mitzva purpose, it retains its kedusha and may no longer be used for general use.
The Shulchan Aruch (OC 42:3) paskens that bags that were set aside to be used for tefillin may not be used for money if they were used even once for tefillin. Such bags are considered to be tashmishei kedusha, items that serve kedusha objects. The Rema adds that one may use the bag for other purposes if they stipulated such before using it.
The Mishna Berura (402:11) notes that this restriction does not apply to a tallis bag, even if one usually places their tefillin bag inside it. Likewise, he writes (Biur Halacha 34:4) that providing that one is meticulous to always put their tefillin back into their boxes before placing them into the tefillin bag, the bag itself does not become tashmishei kedusha and can be used to carry other items.
The Minchas Elazar (1:27) questions whether the tefillin bag still acts as tashmishei kedusha as the knots of the tefillin are not encased along with the tefillin boxes (See Piskei Teshuvos 42:3).
Nonetheless, R' Shlomo Zalman Auerbach (Halichos Shlomo, Tefilla 4:35) maintains that tefillin bags do not require geniza (burial) as tefillin are always wrapped in boxes before being placed into the bag.
In conclusion, one may use an old tefillin bag to store one’s shofar in.