A unital $C^*$ algebra $A$ is said a Tietze algebra if it satisfies the following:

For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi}:C[0,1] \to A$ with $\phi=q\circ \tilde{\phi}$ where $q:A\to A/I$ is the quotion map.

Obviousely every commutative algebra is a Tietze algebra. But what is an example of non Tietze algebra?
Furthermore: Is the familly of Tietze algebras closed under the minimal or Maximal $C^*$ tensor product?