Down syndrome: people with Down syndrome have increased autoimmune susceptibility, including coeliac disease. This is likely related to immune dysregulation and genetic predisposition, so screening for coeliac disease is often considered when symptoms or poor growth are present.

Type 1 diabetes mellitus: both are strongly associated autoimmune diseases and commonly share HLA-DQ2/DQ8 risk alleles. T-cell mediated autoimmunity targets beta cells in type 1 diabetes and gluten-related intestinal antigens in coeliac disease. Hence, people with type 1 diabetes are screened for coeliac disease. NICE evidence review describes the increased prevalence of coeliac and thyroid disease in children with type 1 diabetes.

IgA nephropathy: coeliac disease causes persistent mucosal immune stimulation and altered handling of IgA. This may increase production of abnormal galactose-deficient IgA1 and IgA-containing immune complexes, which can deposit in the glomerular mesangium. The association exists, but it is less direct and less consistent than the type 1 diabetes association. A recent systematic review examined increased kidney-disease risk in coeliac disease.

Selective IgA deficiency: this is a shared immune/genetic predisposition. Selective IgA deficiency is more frequent in people with coeliac disease. Practical exam point: IgA anti-tTG and IgA anti-endomysial antibody tests can be falsely negative in IgA deficiency, so order total serum IgA and, if low, use IgG-based tests such as IgG deamidated gliadin peptide or IgG-tTG.
Scott-Brown's Otorhinolaryngology Head & Neck Surgery, Vol 1, lines 3319-3330 notes the increased incidence and the limitation of IgA serology.

Hypersensitivity pneumonitis / bird-fancier's lung: this is an immune-mediated lung reaction to inhaled antigens, often avian proteins. The reported association probably reflects a general tendency toward exaggerated immune responses and shared susceptibility, but this is not a major routine screening association and does not mean coeliac disease causes bird-fancier's lung.

Lymphocytic colitis: both are immune-mediated gastrointestinal disorders with increased intraepithelial lymphocytes. They can coexist because of shared HLA/autoimmune susceptibility. Clinically, think of it when a person with treated coeliac disease has persistent chronic watery non-bloody diarrhoea, especially if villi have recovered.
Bailey and Love's Short Practice of Surgery, 28th ed., lines 8668-8671 notes the similar inflammatory pattern.

Cerebral calcification with epilepsy: this is the rare coeliac disease-epilepsy-cerebral calcification syndrome, often with bilateral occipital calcifications and occipital seizures. The mechanism is uncertain. Proposed explanations include shared HLA-DQ2/DQ8 genetic susceptibility, immune-mediated CNS injury, and possibly long-standing malabsorption-related nutrient deficiency. It is best thought of as a rare syndrome, not a common complication. A review of the syndrome states that whether it is genetic, due to untreated coeliac disease, or both remains unresolved.