Yes, it is a problem because there is sampling dependence in the responses which needed to be accounted for (although sometimes the effect might be negligible and we do violate assumption all the time when we perform statistical analyses). There are methods to deal with this and, one approach is to include the covariances between related experiments (off-diagonal blocks) in the error variance-covariance matrix (see e.g. Hedges et al., 2010). Fortunately with log ratios this is rather easy. You can get approximated covariances between experiments because the variance (var) of log R is (if Yx and Y0 are independent groups): log Yx - log Y0, to follow the notation in the question, Yx referring to the experimental group and Y0 the control group. The covariance (cov) between two values (e.g. treatment 1 och treatment 2) for log R is cov(loge Yx_1 - log Y0, log Yx_2 - log Y0), which equals var(log Y0), and is calculated as the SD_Y0/(n_Y0 * Y0), where SD_Y0 is the standard deviation of Y0, n_Y0 is the sample size in the control treatment, and Y0 is the value in the control treatment. Now we can plug-in the whole variance-covariance matrix into our model instead of only using the variances (ei) which is the classic way to perform a meta-analysis. An example of this can be found in Limpens et al. 2011Limpens et al. 2011 using the metahdep package in R (on bioconductor), or Stevens and Taylor 2009Stevens and Taylor 2009 for Hedge´s D.
If you want to keep it very simple, I would be tempted to ignore the problem and try to evaluate the effect of sampling dependence (e.g. how many treatments are there within studies? how do the results change if I only use one treatment? etc) .