Does taking oral collagen supplements cause any harm if one has celiac disease?

Probably not: It's also called collagen hydrolysate, collagen peptide, gelatine, gelatine. After 4 weeks of follow-up treatment, a statistically significantly higher skin elasticity level was determined in elderly women. No side effects were noted throughout the study. Reference: http://www.ncbi.nlm.nih.gov/pubmed/23949208

Not if GF: Supplements can be made with binders that can contain gluten. Check the label. The collagen will not affect the celiac.