To check if they're not just artifacts of sequence assembly, you can just do a Southern blot. To see if they're actually transcribed, try qPCR/RNAseq/Northern blot (assuming there's any sequence divergence). If there's no sequence difference, then it's a lot more work.

In all cases, dpryan79 is right to emphasize validation: this is always necessary and important.

From a bioinformatics standpoint, you can tell your aligner to not allow ambiguous alignments. If there are identical regions in multiple genes, these will be ignored and counted as unaligned reads. Truth be told, I don't typically worry too much about this, especially if you have 100+ bp reads.

For variant calling, you'll want to remove true duplicates from the alignment to avoid over-emphasizing sequencing errors amplified via PCR. However, I don't remove duplicates from RNA-Seq data because 1) highly expressed genes will have genuine duplicate reads due to very high coverage (say, >100x for 100 bp reads) and 2) I typically only care about expression levels and possibly splicing events anyways (not variant calling).

genome-walking PCR