Negative efficiency?

I would be inclined in this case to base it upon what the body is actually doing rather than the net movement. On the downward part of the push-up, the muscles are still supporting the weight of the body, but to a lesser degree than during lifting. That is to impart a far lower earthward acceleration than would be caused by un-resisted gravity.
The actual efficiency starts with the initial chemical conversion process of digestion and oxygen metabolism, ATP production and use... all of that stuff at every level of cellular activity, not just in the muscles themselves.

 

I think you are mixing up things here. The mechanical efficency cannot really be compared to the chemical efficency. The efficency of the pushups is more like energy to do pushups/total energy expended.

 

Hi adrenaline,
In mechanical engineering, the efficiency of a pump or compressor is also known as the isentropic efficiency which is equal to the amount of energy an isentropic process would require to compress or pump some fluid, divided by the actual amount of work required. (isentropic means the process has no change in entropy)

Similarly, the efficiency of an expander is also known as isentropic efficiency which is equal to the amount of energy or work that is removed from a fluid by an expander, divided by the amount of work removed if the process were isentropic.

Here's a reference that may help: http://www.taftan.com/thermodynamics/ISENEFF.HTM

There are other types of efficiency in engineering, but I think these are the most applicable to what you're asking. They apply to a mechanism and its thermodynamic efficiency.

The point is that in order to calculate efficiency, you need a definition. There is nothing intrinsic about the two definitions given above, they are simply definitions that we can all use and makes the most sense. It doesn't mean there really is such a thing as an absolute efficiency. For example, an isothermal compression process will require less work than an isentropic one, so such a compressor would actually have an isentropic efficiency greater than 100%! If a compressor ended up putting a lower pressure out than the pressure in, the isentropic efficiency would be less than 0%. It all depends on how you define efficiency.

So to answer your question, I think it doesn't make sense to ask what the efficiency of a person doing a pushup is. One needs to define efficiency in order to measure it and give it any quantifyable value. Depending on how you do that, efficiency can be anything if the process isn't somehow intrinsic to physics.

Side note: some processes are intrinsic, such as the overall increase in entropy in the universe given any process, but I don't know that really applies here.