Context. 55 Cnc e is a transiting super-Earth orbiting a solar-like star with an orbital period of ~17.7 h. In 2011, using the Microvariability and Oscillations in Stars (MOST) space telescope, a quasi-sinusoidal modulation in flux was detected with the same period as the planetary orbit. The amplitude of this modulation was too large to be explained as the change in light reflected or emitted by the planet. Aims. The MOST telescope continued to observe 55 Cnc e for a few weeks per year over five years (from 2011 to 2015), covering 143 individual transits. This paper presents the analysis of the observed phase modulation throughout these observations and a search for the secondary eclipse of the planet. Methods. The most important source of systematic noise in MOST data is due to stray-light reflected from the Earth, which is modulated with both the orbital period of the satellite (101.4 min) and the Earth’s rotation period. We present a new technique to deal with this source of noise, which we combined with standard detrending procedures for MOST data. We then performed Markov chain Monte Carlo analyses of the detrended light curves, modeling the planetary transit and phase modulation. Results. We find phase modulations similar to those seen in 2011 in most of the subsequent years; however, the amplitude and phase of maximum light are seen to vary, from year to year, from 113 to 28 ppm and from 0.1 to 3.8 rad. The secondary eclipse is not detected, but we constrain the geometric albedo of the planet to less than 0.47 (2σ). Conclusions. While we cannot identify a single origin of the observed optical modulation, we propose a few possible scenarios. Those include star-planet interaction, such as coronal rains and spots rotating with the motion of the planet along its orbit, or the presence of a transiting circumstellar torus of dust. However, a detailed interpretation of these observations is limited by their photometric precision. Additional observations at optical wavelengths could measure the variations at higher precision, contribute to uncovering the underlying physical processes, and measure or improve the upper limit on the albedo of the planet.