Not just for a night or two, but forever. For example, if I decided to not sleep ever, what would happen? Would I go crazy? Would I die?

You might experience something like the following:

24 hours -- Mild delirium, mood swings and periods of intense sleepiness or activity
48 hours -- Impaired coordination, hormonal changes and decreased general memory function, but improved short-term memory (according to at least one study)
72 hours -- Visual hallucinations and unintentional periods of "microsleep" (a few seconds to a minute of blacking out)
100+ hours -- Re-enactment of Fight Club
11+ days -- Win the prize for longest period a human has voluntarily gone without sleep (current holder is one Randy Gardner)
2+ weeks -- Some chance of death. Experiments with rats showed that they died after 2-4 weeks of researchers keeping them awake in an experimental setting, probably from hypermetabolism
6+ months -- Certain death; staying awake this long is only associated with rare disorders like Fatal familial insomnia



An interesting detail is that studies of various humans and animals going for long periods without sleep don't show significant short or long term health effects -- when you eventually do die from lack of sleep, it's probably not from a nasty cold that got out of control.

Anecdotally, not getting enough sleep (as opposed to getting none at all) does significantly increase your chance of getting sick, but I don't see any research on that. Perhaps the sickness thing is related to stress.

Source: https://www.quora.com/What-would-happen-if-a-human-did-not-sleep-at-all
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Even 1 hour of delay in sleep makes my next day sleepy especially in working hours.

This is an impossibility. The nature won'r allow it.