PG1995, you need to develop an experimentally inquisitive mind. The answer to your question is in fact something you can measure yourself. To a physicist, the answer is "Yes, it will always consume some power". The reason is you're putting matter across a voltage and, regardless of the resistivities of the materials involved, some current will flow and, unless the resistivities are zero, this current flow will result in increased jiggling of the atoms of the matter (it only takes one electron scattering event to make this a true statement). Whether it's measurable or not is a separate issue.
Your job as experimentalist is to then see if the data support this statement or not. Of course, if the matter you put across the voltage was, say, pure quartz, that current flow is going to be hard to measure. But, in the practical case of a wall wart, it will likely be made of conductors, maybe a transformer, and some active devices, so there will be current flow. The question is: how much?