If sleeping is related to 'saving information' you've experience, is forgetting related to more energy? So I was watching this youtube video (Only check 1:05 - 1:25) where it suggested we sleep because the brain needs rest to save/process all information we perceived throughout the day. I directly made the connection to my own body with the following thought:


  I can stay up for extended periods without the need of sleep. However, I easily and very quickly forget anything, almost like dementia, but if I try hard, I can remember it.


If the necessity of sleeping is related to the amount of information received, does forgetting information reduces that necessity? And does forgetting result in longer days without being tired?

Keep the jargon and technical details to a human level, please.
Short answer: No.

Long answer: The need for sleep is not a function of information received in any meaningful sense. Memory consolidation and forgetting processes are not thought of in terms of energy expense and conservation by any current cognitive science models, and lack of sleep will not necessarily cause you to go mad or die as such. Both are active processes; there's nothing inherently more energy-conserving about forgetting a memory than consolidating one, or vice versa.

Memory consolidation and forgetting processes in the brain are both, ultimately, physical changes like any other, which implies that energy must be spent in some sense. This energy is simply a thermodynamic quantity that has virtually nothing to do with how tired we feel or mental energy, however. Cognitively speaking, energy isn't a well-defined or scientifically meaningful term, so it's not particularly surprising that it ended up on a list of "five simple questions that science can't answer." 

For an example of an active inhibitory forgetting mechanism with references, see my question about retrieval-induced forgetting. If you want to know about memory and the study of memory, I recommend reading Baddeley, Eysenck and Anderson's comprehensive Memory (2009). While it does not directly deal with sleep in depth, it will equip you to ask the right questions about how sleep relates to memory and learning.