Why does humidity cause a feeling of hotness?

When the ambient humidity is high, the effectiveness of evaporation over the skin is reduced, so the body's ability to get rid of excess heat decreases.

Human beings regulate their body temperature quite effectively by evaporation, even when we are not sweating, thanks to our naked skin. (This, supposedly, is also what made it possible for early hominids to become hunters by virtue of being effective long-distance runners.)

Humans are so good at this, we can survive in environments that are significantly hotter than our body temperature (e.g., desert climates with temperatures in the mid-40s degrees Celsius) so long as the humidity remains low and we are adequately hydrated. (Incidentally, this is also why we are more likely to survive being locked in a hot car on a summer day than our furry pets.) In contrast, when the humidity is very high, even temperatures that are still several degrees below normal body temperature can be deadly already.

It because the higher humidity makes it more difficult to cool the body. Even though the room temperature is below body temperature, you generate more heat than needed to maintain your body temperature. To help cool you down, you sweat, and the water evaporates from your skin. The evaporation of the water cools you down.

When the humidity is higher, the sweat does not evaporate as fast, so your skin temperature goes up. This makes you think the temperature is higher when the humidity is higher.

For the thought experiment in your question, the answer is: it depends on the temperature of the room!

Cold Room

If the room is cold, humidity will actually make it feel colder! This is because water vapor has a much higher heat capacity than dry air, meaning that it takes more heat to raise or lower its temperature. So a volume of air with a lot of water vapor can transfer more heat to (or from) your skin than the same volume of dry air.

Warm Room

If the room is moderately warm (around 27–37 °C, I think), it's close enough to your own body temperature that radiative heat transfer is minimal, so the main difference in perceived heat is due to the water vapor's effect on your body's ability to use evaporation (of sweat) to cool itself. This is what the other answers are talking about.

Hot Room

If the room is hot, heat capacity again plays the dominant role. A given volume of high-humidity air contains much more heat than an equal volume of dry air, so it is able to transfer a lot more heat to your body. This is why dry saunas are frequently operated in excess of 100 °C, while a steam sauna at that temperature would be instantly scalding!

In a hot room there is also the issue of condensation. If your skin is at or below the dew point, water will condense out of the air onto you. Condensation is an exothermic phase change, so it will transfer additional heat to your skin.