All internal energy such as thermal, rotational, and internal potential energy contributes to the rest mass of an object. So, yes, a hot object has greater rest mass and would weigh more when measured, if a scale were sensitive enough.

Scales measure the force of attraction between you and the Earth. This force of attraction between you and the Earth (or any other planet) is called your weight. If you are in a spaceship far between the stars and you put a scale underneath you, the scale would read zero. Your weight is zero.

There is less gravitational pull in high places, so you would weigh a tiny bit less on the top of Mt. This is because the peak of a mountain is higher and farther away from the center of the earth, so the pull of gravity in that spot is weaker than at the bottom of the mountain.

If you have absolutely identical objects that have the same weight exactly when they are at the same temperature, then when one object is heated, it will weigh more. This is because the gravitational force depends on the stress energy tensor in general relativity.

Air rises when warmed by a hot sample. This creates convection currents that cause the mass displayed to be unreliable. Wait for the sample to cool to room temperature before weighing it.

Allow the evaporating dish to COOL. Weighing a hot dish will transfer the heat through the metal pan of the balance and ruin the electronic balances circuitry.

The empty crucible is heated to remove moisture and ensure that the crucible is completely dry as any water present may affect the reading. If you skip this step, the crucible, when weighed may include something that will evaporate or combust, changing the weight but not having anything to do with your experiment.

Do not touch the crucible with your hands (oils contaminate it and/or you could be severely burned). Do not place a hot crucible on a lab bench (the temperature difference may cause it to break).