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Here's my take:

請求 is like a compensation demand which is naturally expected to be satisfied or maybe even based on the law. Most examples feature bills payments.

要求 is more like a strong demand from the other party without any implications, like a pay raise demand.

But what about the last one 需要? From the sample sentences I conclude this is a social/market demand, so the antonym in English would be "supplies".

Please correct or confirm my guess.

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請求 is a request for an item for something they have a right to have, like information or money (as a transaction).

要求 is a demand for something, or a demand for someone to do something. Unlike 請求, whether it is actually deserved or not is out of the question.

需要 is the economic demand for something. If the number of sales of cars goes up dramatically, you can say the 需要 for cars increased.

To illustrate the differences by example, imagine you are in a convenience store.

  • If they are very busy today with lots of customers, the 需要 for their products is high.
  • If you purchase something, they will 請求 a payment for the thing you bought.
  • A robber enters and will 要求 the cashier for the money in the cash register.
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    Easy to remember: 需要 (demand) goes along with 供給 (supply) to form 需給 for "supply & demand" (although note the order in Japanese actually makes it "demand & supply"). – istrasci Jun 15 '18 at 15:32

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