Let $f$ be a classical modular form of weight $k \geq 2$ and $M_f$ be the motive attached to it. If $M_f$ is a CM motive then is it true that $f$ has CM? For weight $2$, I believe this is a result of Shimura. He shows that if the variety $A_f$ has CM then it can be written as a product of several copies of an elliptic curve E with CM by $K$ and $f$ arises from the Hecke character associated to $K$.

The answer is yes. Motives $M$ with CM are determined by algebraic Hecke characters $\chi$. Hence their associated modular forms $f$ have CM. References that contain detailed discussions about this relation include articles by Anderson (1986), and Blasius (1986), as well as Schappacher's contribution to the Motives volume edited by Jannsen, Kleiman and Serre, AMS 1994. 

