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.