Temperature isn't a single string or atom, it is a macroscopic phenomenon.
If a stand alone string/atom moves into a higher energy state it must absorb photon, in order to reach a lower state it must discharge photon. That process does change the mass of an object. However, the differences are nuances, insignificant small.

It would be funny if temperature would make a significant difference in mass and weight.
- As soon as you lighten a match, your hand smashes to the ground, because it suddenly became t0o heavy-