We present the results of an intense photometric monitoring in the near-infrared (~0.9 microns) with the TRAPPIST robotic telescope of the newly discovered binary brown dwarf WISE J104915.57-531906.1, the third closest system to the Sun at a distance of only 2 pc. Our twelve nights of photometric time-series reveal a quasi-periodic (P = 4.87+-0.01 h) variability with a maximal peak-peak amplitude of ~11% and strong night-to-night evolution. We attribute this variability to the rotational modulation of fast-evolving weather patterns in the atmosphere of the coolest component (~T1-type) of the binary, in agreement with the cloud fragmentation mechanism proposed to drive the spectroscopic morphologies of brown dwarfs at the L/T transition. No periodic signal is detected for the hottest component (~L8-type). For both brown dwarfs, our data allow us to firmly discard any unique transit during our observations for planets >= 2 Rearth. For orbital periods smaller than ~9.5 h, transiting planets are excluded down to an Earth-size.