Do fire-taped drywall seams need to be painted?
No, paint is not required for wall board on combustible (wood) construction. (See ICC Table 720.1(2), Items 12, 14, and 15.)
Note: Item 15 includes 2-hour fire rated construction too (two layers of gypsum board).
However, be aware that nailing requirements, type of plaster mix, etc. do matter. Most stud spacing is 16” on center, but I did see one with 24” on center (14-1.4 ) with gypsum board long dimension to stud, with 6d cooler nails or wallboard nails at 7” on center.