Among the 19 red-giant stars belonging to eclipsing binary systems that have been identified in Kepler data, 15 display solar-like oscillations. We study whether the absence of mode detection in the remaining 4 is an observational bias or possibly evidence of mode damping that originates from tidal interactions. A careful analysis of the corresponding Kepler light curves shows that modes with amplitudes that are usually observed in red giants would have been detected if they were present. We observe that mode depletion is strongly associated with short-period systems, in which stellar radii account for 16%-24% of the semi-major axis, and where red-giant surface activity is detected. We suggest that when the rotational and orbital periods synchronize in close binaries, the red-giant component is spun up, so that a dynamo mechanism starts and generates a magnetic field, leading to observable stellar activity. Pressure modes would then be damped as acoustic waves dissipate in these fields.