Cluster cool cores possess networks of line-emitting filaments. These filaments are thought to originate via uplift of cold gas from cluster centers by buoyant active galactic nuclei (AGN) bubbles, or via local thermal instability in the hot intracluster medium (ICM). Therefore, the filaments are either the signatures of AGN feedback or feeding of supermassive black holes. Despite being characterized by very short cooling times, the filaments are significant H$\alpha$ emitters, which suggests that some process continuously powers these structures. Many cool cores host diffuse radio mini halos and AGN injecting radio plasma, suggesting that cosmic rays (CRs) and magnetic fields are present in the ICM. We argue that the excitation of Alfv'en waves by CR streaming, and the replenishment of CR energy via accretion onto the filaments of high plasma-$\beta$ ICM characterized by low CR pressure support, can provide the adequate amount of heating to power and sustain the emission from these filaments. This mechanism does not require the CRs to penetrate the filaments even if the filaments are magnetically isolated from the ambient ICM and it may operate irrespectively of whether the filaments are dredged up from the center or form in situ in the ICM. This picture is qualitatively consistent with non-thermal line ratios seen in the cold filaments. Future X-ray observations of the iron line complex with XARM, Lynx, or Athena could help to test this model by providing constraints on the amount of CRs in the hot plasma that is cooling and accreting onto the filaments.