There is no definite correlation between Hashimoto's thyroiditis and elevated CRP and ESR. However, Hashimoto's disease has been linked to other autoimmune diseases, including SLE, rheumatoid arthritis and Sjögren's syndrome with associated changes with acute phase reactants.
Ref: Erden S, Buyukozturk S, Vural P, Değirmencioğlu S. Acute phase reactants in Hashimoto thyroiditis. Int Immunopharmacol. 2008 Dec 20;8(13-14):1863-5