tl;dr A tiny p value allows you to confidently reject the null hypothesis of no difference. It doesn't really matter much how tiny. Once the FDR is below 0.05, you reject.
Ignoring the complexities of interpreting such a value, it depends. With (eg) R, an architecture specific limit (eg 2.225074e-308 - see https://stat.ethz.ch/R-manual/R-devel/library/base/html/zMachine.html) defines the threshold below which you see a zero. It's a sign that the real "true" value is very, very small.
Assuming you could calculate it correctly using infinite precision arithmetic, the probability that the true value for p from any biological experiment is exactly zero is itself vanishingly small.
What are the raw counts for the contig in the two groups? The raw data is the key here.
If one group's raw counts for the contig are all consistently very low and the other group's are all consistently very high then you may have good grounds for asserting that the groups have different expression.