There are several formulas of Robitussin with various ingredients. The main ingredient in all of them is Guaifenesin, which is an expectorant. Other formulas have different ingredients that should be OK, but warrant asking your doctor or pharmacist about.
For Example, Robitussin Cold and Robitussin PE both contain psuedoephedrine, which is not recommended for hyperthyroid people and, like any other drug, may be metabolized more slowly in hypothyroids so you would need to take a longer time between doses (if it says take every 4-6 hours stay closer to 6).
Which formulation are you askin about?
I am not a doctor, nor have I ever played one on TV...