This question originally appeared on Quora. 
Answer by Frank Heile, Physicist, Software Engineer, Consciousness Theory
Yes. 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. The stress energy tensor 00 component is the total energy of the body, which includes the rest mass plus the kinetic energy of the object. Temperature differences means that there is a different amount of kinetic energy in the motion of the atoms of the two bodies.
For example, if you start with two identical kilograms of water at 0 Celsius, and if you then heat one of them to 100 Celsius, then the kilogram at 100 Celsius would be heavier by an amount equivalent to 4.6 nanograms of additional water weight (see 100*1000 calories / c^2 ).
Now 4.6 nanograms is not very much, but it is equivalent to 154 trillion molecules of water (see 4.6 10^-9 gm water in molecules ). Just imagine - the energy used to heat the water is equivalent to the weight of 154 trillion additional water molecules if they could be converted completely into energy (remember )!
This 4.6 nanograms weight difference is far too small to measure. For example, the most sensitive commercial balance I could find (at $26,000) can compare two weights up to 520 gm with an accuracy of about 0.01 mg - which is off by a factor of about 10,000 compared to what would be needed to measure this effect. So this is a purely theoretical effect which has not been experimentally measured at this level of precision, but there really isn't any doubt that this is what happens.
The heated water will expand compared to the colder water, so it will displace more of the air than the cold water, and the hot water will thus experience a buoyant force. I calculate this effect would make the hot water weigh 33 milligrams lighter than the cold water (see (100 celsius) * (coefficient of thermal expansion of water)*(1 liter) *(density of air)). So this totally swamps the effect of the heat adding 4.6 nanograms of weight to the hot water. To overcome this effect, the weighing would have to be done in a vacuum. Also both containers would have to be sealed so that no water evaporates - evaporation would totally swamp the effect.