Note: This answer is based on events seen up to Chapter 651, and contains unmarked spoilers.
Since the answer is long, I will divide it into several sections, first explaining the mechanism of the Infinite Tsukuyomi, then the background and philosophy, and finally answer the specific questions.
Mechanism of the Infinite Tsukuyomi
Use Itachi's Tsukuyomi as a reference to understand the Infinite Tsukuyomi. The illusion lasts for 72 hours as per the flow of time perceived in the illusory world, but for only a moment in the real world. Powered by the Juubi/Shinju's chakra, Infinite Tsukuyomi is a superpowered version of it, in three ways:
- It draws everyone's minds into the illusion.
- Its illusory space is designed in great detail, as desired by the caster.
In particular, people who do not exist in the real world can be created there.
- It lasts forever as per flow of time perceived in that world.
The illusion appears to last for only a moment in the real world, but since it is endless, real world time doesn't appear to advance. Moreover, since everyone's minds have been drawn into the illusion, nobody is around to perceive the flow of time. This may also be a subtle cultural reference, since Infinite Tsukuyomi uses the moon to cast the illusion, and Tsukuyomi, the Moon God is associated with the flow of time.
To understand the Moon's Eye Plan philosophy, a background of the Naruto story is useful.
Background of the Naruto story
Kishimoto originally wanted to make Naruto a seinen manga, but was advised to make it shonen, so that it could be published in the Weekly Shonen Jump magazine. In my opinion, it still retains its seinen core in the plot, while using shonen elements in the narrative.
Amidst the action and comedy, it has explored various elements of human nature or philosophy, and conflicts between them. The characters in the story are also used to represent such elements. A narrative primarily focused on the philosophy would lose audience interest, so it is useful to have characters represent them. For example, Naruto represents how positive thinking and hard work can overcome adversities, while Sasuke represents how revenge makes a person blind letting others use him, and so on.
Objective of achieving a peaceful world
An overarching philosophical question asked several times is, "How to achieve a peaceful world, devoid of hatred and suffering?" Various characters have attempted to answer it in their own way.
- Hashirama: Distribute powers evenly between villages, so that they won't fight over it.
- Jiraiya: Some day, people will truly understand each other, and hatred will cease. I don't know how it will happen, but I believe in my student to figure it out.
- Nagato: Create a powerful weapon which people will fear and stop fighting. This peace is temporary, since people eventually forget the weapon's power, so they need to be reminded again.
- Naruto: I will end the cycle of revenge and justice, which will lead to peace somehow.
- Madara: Create a perfect world free of winners and losers, which will end hatred and suffering.
Motivation for the Moon's Eye Plan
Madara lost all his brothers in the war. He lost the Hokage title to Hashirama. His own clan rejected him when he wanted to protect them (according to him). He also lost to Hashirama at the Valley of the End. He then read the world history at the Uchiha shrine, which described how Kaguya Ootsutsuki ended all existing conflicts by eating the Shinju's forbidden fruit, but it only led to more conflicts. He realised that a world which creates winners also creates losers, such a world would always be at conflict, and any effort to end the conflict would cause more conflict. He was convinced that there is no hope to achieve peace in the real world, and hence, decided to achieve it in an illusory world.
Obito initially rejected Madara's plan, but seeing how the ninja system led to Rin's death made him lose hope in the real world, and convinced him to accept Madara's plan of creating an illusory peaceful world, where Rin doesn't have to die.
Answers to the specific questions
When everyone is under the genjutsu, does that mean no one is actually
living their own life? Or do people live their lives and the usage of
the genjutsu just eliminates pain, suffering, and war?
Will the people starve to death under this technique? And if not, are
they able to reproduce? And if they can do the above two, will this
technique eventually cease when Madara/Obito die?
I'm just wondering if life can still continue under this technique? If
so, does the technique ever stop?
As explained above, the illusion lasts forever as per time perceived in the illusory world. The real world becomes irrelevant once the illusion starts.
Now on a more (deeper) philosophical level, how is this plan any
different from Pain's total destruction plan?
Pain's plan was to achieve peace in the real world through the fear of a powerful weapon, although the peace is temporary. Madara's Moon's Eye Plan gives people no choice, they are forced into the peaceful world permanently. In my opinion, the Moon's Eye Plan represents the philosophy that peace is a state of mind, while the body is unimportant, since it draws people's minds into peace (albeit an illusory one), while their bodies left in the real world become irrelevant.
As a side note, Pain's plan seems influenced by the atom bomb in our world, though the author hasn't stated it explicitly.
Why bother with the Eye of the Moon plan if you can just kill the entire planet.
The difference between the two could be a long philosophical discussion unto itself, so I will cut it short using an analogy. To end a lung tumour patient's suffering, you could do:
- Remove the tumour (Naruto's solution)
- A lung transplant (Moon's Eye Plan)
- Kill the patient (this question's suggestion)
Analogies are often misleading, but the key point is that the Moon's Eye Plan wants to create a peaceful world, which everyone can experience, which destroying the world cannot achieve.
This then leads me to believe that Madara and Obito just have psychological problems. Is this the reason why they are proceeding with this crazy plan?
There is nothing inherently crazy about wanting to escape the pains of reality in one's dreams, since this is quite common human nature. The following Calvin and Hobbes strip is one popular instance. Bill Watterson's cat, Sprite, died in real life around that time, and he realised that he could be always be with his cat in his dreams.
Concluding that Madara or Obito have psychological problems is a stretch, or subjective at best. In our world, it is quite common for one person to experience a problem, and then try to solve it for everyone. Madara or Obito could easily solve only their problem using Izanagi, Rinne Tensei, or limiting the Tsukuyomi illusion only to themselves.
Lastly, the war also represents the conflict between Madara/Obito's philosophy and Naruto's philosophy, and we know that Naruto would eventually win in some way. This could be Kishimoto's way of giving us another moral, that trying to solve real world's problems through an illusion is futile, and they should be solved in the real world.