Will an MRI show a lumbar herniated disc even after anti-inflammatory medications are given?

Yes. Should not affect the test sensitivity of test at all.
Herniated Disc. Yes, an MRI should still show a herniated disc even after anti-inflammatories have been used. The drugs should help reduce swelling, but a herniated disc is a physical bulge from the disc pushing into the vertebral canal causing pressure on one or more nerve roots. No drugs can change that, even if they make the perceived pain less.