Due to the food chain and the cycle of life, matter is constantly moved between plants, primary consumers, secondary consumers, and so on, until the consumers die, giving matter they consumed back to the ecosystem via dung and corpses. This should mean that there is a constant flow of mass and energy moving through an ecosystem. Does that mean we could (theoretically) weigh a given ecosystem in metric tons?
You can always weigh the biomass of the (global) ecosystem in metric tons, or any other unit, even when this mass would not be constant.
I don't see a reason why the total biospheric mass should be constant (maybe because it reached a saturation limit long ago) on short timescales. On long/geological timescales it is certainly not, as the available carbon, silicate, oxygen etc. for the bottom of the nutritional chains a.k.a. microbes, lichen, fungi... evolves, as the planet itself changes.
That is however not true for the total mass of Earth, which is essentially constant since its time of formation, but the biospheric mass is anyways only a small fraction thereof.