We examined the variability of absorption line strength of intervening systems along the line of sight to GRB 060206 at z = 4.05, utilizing low-resolution optical spectra obtained by the Subaru telescope from six to ten hours after the burst. Strong variabilities of FeII and MgII lines at z = 1.48 during t = 5--8hr have been reported for this GRB, and those have been used to support the idea of clumpy MgII cloudlets, which was originally proposed to explain the anomalously high incidence of MgII absorbers in the GRB spectra compared to quasars. However, our spectra with a higher signal-to-noise ratio do not show any evidence for variability in t = 6--10hr. There is a clear discrepancy between our data and those of Hao et al. (2007, ApJ, 659, L99) in the overlapping time interval. Furthermore, the line strengths in our data are in agreement with those observed at t ˜ 2hr by Thöne et al. (2008, A&A, 489, 37). We also detected FeII and MgII absorption lines for a system at z = 2.26; these lines do not show evidence for variability either. Therefore, we conclude that there is no strong evidence for the variability in the intervening absorption lines toward GRB 060206, offering poor support for the MgII cloudlet hypothesis by the GRB 060206 data.