This work is inspired by the expansion of the plasma bubble in a narrow gap reported by Chu and Lee [Phys. Rev. Lett. 107, 225001 (2011)]. We report the unstable phenomena of the plasma-liquid interface with different curvature in a Hele-Shaw cell. Dielectric barrier discharge is produced in the cell at atmospheric pressure which is partially filled with silicone oil. We show that the Saffman-Taylor-like instability is observed on the bubble-type, channel-type, and drop-type interfaces. The Schlieren observation of the plasma-drop interaction reveals that there is a vapor layer around the drop and the particle image velocimetry shows the liquid flow inside the drop. We propose that the thermal Marangoni effect induced by the plasma heating is responsible for the unstable phenomena of the plasma-liquid interaction. The fluctuation of the interface is shown consistently with the Saffman-Taylor instability modified by the temperature-dependent velocity and surface tension.