Is the universe getting lighter?

What with all the stars burning hydrogen and what have you, is the mass of the universe decreasing with time? Or does electromanetic radiation have the same “weight” as the mass that is destroyed to create it? Also, is it possible to estimate what proportion of the mass that existed shortly after the big bang has, by now, been converted into energy?

Correct. If you convert 1 kg of matter into energy, that energy will also have 1 kg of mass. The Universe is a closed system, by definition, so it masses the same now as it did immediately after the Big Bang.

Briefly, yes. To go into more detail:

Mass can be considered as the portion the energy of a system which can’t be transformed away by going to a different reference frame. So, for instance, if you have a spaceship flying along, it has kinetic energy because it’s moving. But it you’re flying along next to the spaceship at the same speed, in your reference frame it’s not moving, so it doesn’t have kinetic energy. You transformed away the kinetic energy, so that kinetic energy was not mass.

On the other hand, suppose you had two spaceships, moving in opposite directions. Now, you can move along with one spaceship, but if you do, then the other spaceship is moving even faster. By moving alongside one spaceship, the total kinetic energy of the system is actually increased, not decreased. The lowest energy you can get is if you stay in the center of mass frame of the system. So in that case, the kinetic energy can’t be completely transformed away, so some of the kinetic energy is part of the mass of the system.

Now, the only sort of energy light has is kinetic energy: A single photon, or a beam of photons all moving in the same direction, does not have mass. The faster you go alongside that beam, the less energy you’ll measure for the beam. You can’t get the energy all the way down to zero, since that would require you to move at the speed of light, but you can get arbitrarily close. So we say that photons are massless.

On the other hand, if you have photons moving in different directions, then the system as a whole has energy you can’t transform away. If you move to a reference frame where one photon has less energy, some other photon or photons will have more energy. So even though individual photons don’t have mass, a system of many photons can and usually does have mass.

So to sum up, even though stars are giving off light and losing mass, the light the stars give up has all of the mass the star lost. The total system of star + all light given off by the star does not change its total mass, and neither does the Universe as a whole.