There are many studies accounting the negative effects of sleep deprivation which all seem to detail the deficiency of time for your body to "reset" and how this leave it in a depleted state. If you: efficiently supplemented the body with nutrients, also accounting for the extra time awake; rested enough throughout the day for your body to recover fully; maintained a healthy and focused mindset; maybe even adapted/trained your body to prevent the negative effects, could you go without sleep?
Nutrients have little to do with it*, it's the rest cycles that are important.
Cells are messy, stumbling, statistically functional things. Periodically, a cell needs to take a break so it can perform basic maintenance tasks and undo the disorder its incurred handling numerous signaling and protein construction mechanisms . For neurons, this is particularly difficult. If they take a break, they begin to lose information as they undergo long term changes in response to their activity level. So instead, animals with sufficiently complex nervous systems enter a state of reduced activity (called sleep) to help perform maintenance  - there's also some question to how it might contribute to memory, but that subject is more contentious .
There may be volitional mental states that are achievable that can do the same things as sleep can (give all the different parts of the nervous system rest cycles and slow down metabolism) but nobody has been able to prove it yet, that I know of. Harvard researcher, Dr. Herbert Benson, claimed it was possible in The Relaxation Response, but I don't know of any one actually demonstrating it in trials.
-  https://www.ncbi.nlm.nih.gov/pubmed/25325492
-  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3972489/
-  http://www.sciencedirect.com/science/article/pii/S0896627304005653
*Granted, if you don't sleep, your metabolism doesn't slow down and you burn through nutrients faster.
2$\begingroup$ Hi Keegan. Do you have any references to give us for something to read more about the statements you have made in your answers? $\endgroup$ Jul 10, 2017 at 4:30
1$\begingroup$ citations added $\endgroup$ Jul 14, 2017 at 1:39