The question why rich countries should help poor countries is based on hunanitarian principles.

我觉得可以把why 改为that！

我学校的内部语法书上在讲表语从句时有这么两句：

The reason why (that) he has been such a success is that he never gives up. 

他获得如此成功的原因是由于他从不放弃。

The reason why (that)  he did not come to the meeting is that he was ill．

他没有参加会议的原因是由于他病了。(常用Ther reason is．．．that，不用because)

参考译文：

为什么富国应该帮助穷国的问题是基于人道主义原则的。

that引导陈述句，why引导疑问句。既然是question的同位语，从句就必然是个问句。用that引导问句就是错误的。

不能，如果你用that的话，变成陈述内容了，question就失去了疑问含义了，就不成question了。

      why rich countries should help poor countries 是 The question 的同位语从句，why 不能改为that。例如：

      Your original question, why he did not report it to the police earlier, has not yet been answered.  你提出的问题，他为什么没有早向警方报告，还未得到答复。(摘自赵俊英《现代英语语法》p.1238)