Homework : Sodium Potassium pump
I can't yet comment, so here's a (sort of) answer:

I've only ever heard the "2/3 of cellular expenditure" rule applied to neurons, and frankly, it wouldn't make much sense in any cell other than a neuron.  The Guyton and Hall Textbook of Medical Physiology says, "For some cells, such as electrically active nerve cells, 60 to 70 percent of the cells' energy requirement may be devoted to pumping Na+ out of the cell and K+ into the cell."  A cardiac muscle cell, for example, would probably use more of its energy for contraction than pumping ions.

In a neuron, the bulk of energy expenditures are for electrical signalling done via allowing sodium and potassium ions to temporarily run DOWN their respective gradients.  As such, the "purpose" of a neuron is to pump ions across a membrane and then let them run back when needed.  Since a neuron doesn't really DO much else, it makes sense that the bulk of the energy produced will go towards pumping ions.  A practical example is the use of barbiturate comas to reduce metabolic activity in the brain.  Flat-lining the EEG drops metabolic requirements by ~50% (Source: Medscape).

Please keep in mind that this answer is specific to neurons...other cells perform other functions, and those other functions will take up significant fractions of energy produced.  Further, other cells probably aren't devoting large fractions of their energy towards signalling, and so the fraction of energy devoted to pumping ions will be markedly less.  They'll still be pumping ions to maintain a gradient to perform tasks like membrane transport and osmotic control, but those tasks are ancillary to the "purpose" of those cells.  I can't find any solid numbers for Na/K ATPase energy expenditures in non-neuronal tissue at the moment, but if I do, I'll update appropriately.

UPDATE:
With regards to the question itself, which I apparently forgot to answer:

Can a cell function without an appropriate membrane potential? Can co-transport run without a membrane potential?