I think that in most situations, the room will heat up initially.
However, if you have low humidity, and spread out the water quickly enough so that surface area of the water is big, the evaporation of the water will actually require more energy than the latent heat energy that exist in the boiling water and the initial warm up is diminished.
If the humidity is 100% at room temperature, water will still evaporate from the 100c degree water as it has a higher vacuum pressure. However, if the room is airtight, the water will just condensate inside the room, but there will not be any cooling effect from any further evaporation after the room has reached thermal equilibrium as the room will remain at 100% humidity but at a higher temperature.
If you have an airtight room at 20 degrees Celsius with 0% humidity, each cubic meter of air can hold up to 17.3 grams of water moisture.
1 kg(liter) of 100 degree water holds enough latent heat energy to evaporate 148 grams of water in a 20 degree warm room.
So if I got this correct, the room needs to be at least 8.55 cubic meters big to counteract the initial heating from the evaporating water and start cooling off from the rest of the water evaporating.
https://www.engineeringtoolbox.com/maximum-moisture-content-air-d_1403.htmlhttps://en.wikipedia.org/wiki/Latent_heat#/media/File:Water_temperature_vs_heat_added.svg
