Dealing with poor quality RNA is never ideal. If it is possible to re-run the experiments (with larger N so that your power doesn't go super low if a few samples fail), that is always the best solution. That being said, sometimes this is not possible.
What type of gene expression analysis are you trying to do? If you are doing qPCR, you can try running it on your poor quality samples and see if your reaction quality is good by looking at melt curves and the amplification of your housekeeping genes. If you want to conduct RNAseq, while poor-quality RNA is far from ideal, there are methods developed to deal with degraded RNA. Illumina's TruSeq RNA Exome reagents, for example, are designed to do library prep from degraded RNA for sequencing purposes.
I too experienced the same issue (low rin) for my mouse samples few years ago. What we found out was that the expression of the housekeeping genes varied a lot in these compared to samples with high rin where the values did not change much. Gene expression from these low rin samples are not reliable. You might be better off running new set of animal experiments.
Yes, I agree that poor quality samples may result in a low reliability of data. Maybe for some kinds of RNAs this would not matter too much (e.g., miRNAs), but you cannot say generally that their expression is not altered as well. The poor RIN values indicate that sampling is not standardized, RNases are active and may degrade the samples in course of time. Therefore, stabilization using proper reagents (one of the most used is RNAlater) within sampling would be highly recommended in future experiments.
I really agree with all previous opinions, even with qPCR, it was found that RNA can degrade at different levels (depending on many factors) which evantually causes different HKG/GOI expression ratios and thereafter, false differences between good and bad samples.
re-extraction is extremely recommended if you have other well preserved samples