Is it ever okay to have exposed fiberglass insulation in a room?
This room is a hot exercise room and there are open-grate ceiling panels (a few open) where you can see the insulation.
Properly installed fiberglass batt insulation requires a minimum of one layer of poly (plastic sheet) moisture barrier installed at the warm side of the insulation (inside the house). This will ensure no moisture can degrade the insulation and is required by code.
The exposed insulation could also be a cause for indoor air quality concerns if the material becomes airborne. However, it is not considered to pose a danger with respect to fire.
The fact you have mentioned it is a hot room may point to another issue. That is whether you have an adequate amount of insulation at the ceiling!