Can I compare Shannon indices of metagenome gene data?
I'm not an expert on Shannon-Weaver Index, but according to wikipedia it is the same as exponentially transformed Renyi entropy. If it is the case, you can compare them since they are scale invariant summary statistics. If you want error bars, you can always try resampling methods such as bootstrapping. Hypothesis testing can also be done with bootstrapping, although the power of the statistical test may not be very strong and fail to reject the null.