Not exactly, the diversity of your samples is independent of sequencing depth, however, sequencing depth can create a bias in the sequences you can recover, especially if you are interested in microdiversity (those microbes that <1%).
Let’s take the case you have a sample with “regular” diversity in which the microbes have similar abundance. In this case, your sequencing depth won’t affect the diversity you detect because all markers will be sequenced approx equally.
In another case, you could a sample with low diversity in which a microbe has a really high rel. abundance (<70% or similar). If you used a “normal” sequencing depth then you might not be able to detect those very low abundant microbes just for the fact that the chances of sequencing their gene markers are low because of their lo r.a.
Now if would increase the sequencing depth of that sample in order to better detect those low abundance microbes, you might find that some of the gene markers for the high abundance microbes were sequenced twice, artificially increasing the rel. abundance of that microbe. In this case subsampling your sequences could be of use, however, this is hard to detect before hand unless you have previous information on your samples (i.e. amplicon sequencing, or high duplication ratios during your quality analysis). This workshop video has some info on this around minute 11.