I never understood why a hot cup of tea is heavier than a cold one.

Because mass is associated with energy (E=mc2), in a hot cup more energy means more mass.

I don’t understand how trembling molecules can affect mass. After all, in a cup of tea, energy does not turn into mass.

The school teaches that E=mc2.
In the case of a cup, “hot” is energy (E), and “heavier” is mass (m).

My question may seem quite silly, but why is it that the mass increases and not the speed of light in the vacuum?

Because it is a constant.

I understand what you want to say. But this is not obvious from the formula.

“May tea is so strong that even photons accelerate!”