no. It says that mass and energy are equal but they're not the same.

For example, if an object has the mass m and the temperature T.

The energy of both together would be:
E(T) (thermal energy) + E(m) (mass expressed as energy, which theoretically is true because of E = m * c²)

Heating up this object would cause an increase in thermal energy E(T) and thus the total amount of energy increases aswell. But it's not gonna change anything at it's mass.

Although I'm pretty sure that this statement is wrong: you could see mass as a different type of energy, like kinetic, thermal, electrostatic, etc.
Just because the amount of one of these energies increases doesn't mean that the others increase aswell.