Is re-normalization of RNAseq data recommended for analysis of gene subsets?
FPKM, RPKM are ways of normalising for both sequencing depth and gene length. You first calculate a scaling factor (SF = #read/1Mil),you then divide the genes counts for the SF(RPM = # counts/SF), and finally you divide by the length of the gene (RPM/length(gene)). Each sample has a different SF meaning that you should not compare values from different samples. 

To fix this, you have to normalise across samples, by upper quartile normalise using a subset of genes. In this case, the SF is calculated based on the upper quartile (70%) of this shared and immutable subset, and it is therefore the same for all the samples.

If you subset your genes, and you are using the normalised values for downstream analyses, then you need to re-scale your values. 

I personally try to avoid FPKM,RPKM,TPM (if I can) and always start from raw counts. 

I would suggest to have a look at the documentation of packages like DESeq2 and edgeR for detailed and better explanations.