Is the beta-thalassemia gene recessive or dominant?

Neither. People normally have 2 beta globulin genes. A mutation in one or both genes leads to 3 levels of beta thalassemia. The genetics of the disorder is not all-or-none nor dominant-recessive. Instead, more or worse mutations causes worse disease. One mutation = beta thal trait (mild symptoms), 2 mutations = beta thal intermedia (transfusions sometimes) or beta thal major (lifelong transfusions needed).
Recessive. Autosomal recessive typically. Autosomal dominant inheritance has bee reported in some families.