One way to get an idea about possible false negative rate is to conduct an RNA-seq power analysis. There are a number of Bioconductor packages for this. There are various dimensions and parameters at play, including number of samples, sequencing depth, biological variability across replicates, and of course the true effect size.
No, it isn't possible to estimate the FNR for real data. Computing the FNR is only possible in a simulation study or with artificially constructed data for which you know all the TPs.
You could use the propTrueNull() function of the statmod package to estimate the total number of null hypotheses that are actually true, and that would give you some idea of now many false nulls have failed to reach your significance cutoff. That does rely on the p-values being uniformly distributed for truly non-DE genes, which will never be exactly true.
As Michael has said, you can get a general idea from a power analysis of what FNR might be typical for an experiment like yours, given assumptions about effect size etc, but that will not yield an estimate of FNR for any specific dataset.