Statement From The International Sauna Association:
Although sauna may help to prevent the common cold and flu, it’s a different story when you are actually sick with a respiratory illness. In spite of the fact that the temperature in a traditional sauna is basically high enough to kill most viruses, during coughing and sneezing the transfer of a virus from one person into another may be rapid enough to transmit the disease even in the hot room. In washing, resting and dressing rooms the possibility to get an infection is similar to that of indoors generally.
(There is little or no danger of infecting other bathers because heat above 56C (132F) generally kills most viruses. However, high sauna/banya temperatures (in the range of 150-200 F and above) with low humidity (less than 30%), irritate the sensitive linings of the lungs and may actually be detrimental to healing. Lower temperatures and higher humidity (as found in a steam bath or hammam however, seem to provide relief).
So do not go to the sauna/banya if you are sick.