Sauna or steam room when you're sick
Two different questions get asked together here, and they have different answers. Steam for a blocked nose is a reasonable idea and people have done it forever. Sitting in a 90°C sauna while running a fever is not.
Here is the practical distinction, and the point where you should stay out of both.
Steam and congestion
Warm saturated air moistens nasal passages and can loosen mucus, which is why a steam room, or simply a hot shower with the door shut, often feels good with a head cold. It does not shorten the illness and there is no strong evidence it does, but symptomatic relief is a legitimate reason to do something.
Keep it short. Ten minutes is plenty when you are unwell, and the usual advice about hydration matters more than normal because you are likely already running low on fluids.
When to stay out of both
If you have a fever, do not use either. Your body is already running its temperature up deliberately and adding external heat load makes it harder to regulate. Wait until your temperature is back to normal.
Also stay out if you are vomiting or have had diarrhoea, because you are already dehydrated. And stay out if you feel dizzy, faint or unusually weak — heat will make all three worse, and fainting in a locked wet room is the specific scenario that turns a minor illness into a real problem.
Sauna with a cold
A mild head cold without fever is probably fine for a short, cooler-than-usual session if you feel up to it. The dry air will not help congestion the way steam does, and some people find it irritating.
There is a persistent belief that you can sweat out an illness. You cannot. What you can do is dehydrate yourself while your immune system is already busy, which is the opposite of helpful.
Shared facilities and other people
Worth stating plainly: if you are contagious, do not use a gym or hotel sauna or steam room. Warm humid surfaces are not a good place to leave a respiratory virus, and the people in there with you did not choose it.
This is one of the underrated arguments for a home installation. Being able to use a steam room while you have a cold, without being inconsiderate, is a genuine benefit of not sharing one.
If you have an ongoing condition
Asthma, heart conditions, low blood pressure, pregnancy, and medications affecting blood pressure or fluid balance all change this calculation, and they change it more when you are unwell than when you are healthy.
None of that is a reason to avoid heat bathing generally. It is a reason to ask a doctor about your specific situation rather than taking guidance from a builder's website, including this one.
Having your own room changes this
Most of the awkwardness in this question disappears when the room is yours. Steam room installation or a steam shower conversion both give you somewhere to sit with a blocked nose without being inconsiderate to anyone else.
Steam shower conversions are the cheapest permanent route if you already have an enclosure that can be sealed. We also do sauna room installation where dry heat is what the household actually wants, and sauna and steam room installation in Dubai as a pair where the budget allows.