Do Transformers Produce a Load?
It depends what you mean...

They produce a reactive load, which is to say electricity flows through them, but they don't consume energy.

There's also losses which are usually quite small and are a real load
and consume energy to make a small amount of heat, and that humming sound, etc.

If you connect a load on the output side of the transfomer, it appears electrically as-if there was a transformed version of the load at the input side.