If all the steam from a bowl of boiling water is collected, after all the water boils away, would the steam's temperature go up?
I'm not quite able to understand what the question is. Are you collecting water separately as it evaporates from the bowl, or is the steam collected in the bowl which is a sealed one?
In the former case: When you collect the steam as it is produced with the water in the bowl boiling, there will be no change in the temperature of the steam whether it is collected at the beginning or is the last bit of steam collected when the water is getting over.
In the latter case: When the steam accumulates in the sealed bowl, as you continue to heat the bowl, more of the water will turn to steam but the steam produced increases the pressure in the container. This results in an increase in the temperature at which water turns to steam. As heat is introduced into the container the temperature of everything in the container, the water and the steam, increases. The amount of steam and the amount of water present in the sealed container is in a proportion which maintains equilibrium between temperature and volume. The temperature of the steam here will go up as more heat is introduced.
If water is heated to boiling the vapor coming off the surface of the liquid water is steam which is at the same temperature as the boiling water. At sea level, that temperature is 100 degrees C or 212 degrees F. As long as the pressure does not change, the temperature of the steam will remain at 100 degrees C.
By collecting the steam and pressurizing it you can increase its temperature to a much higher level. This is done in city heating systems, power generation, and other applications. Once the steam is generated by heating the water, it can then be pressurized and heated to a higher temperature, without any liquid water being present.