We all feel___our duty to help those people in need.A.that B.this C.it D.what

C
形式宾语

C此处it是用作形式宾语，代替后面的动词不定式，不能用that.

A.that 引导宾语从句

C

A
固定搭配,that是个连词
好象还可以用it's

选C。这里有feel就不能it's了。如果要选A的话，后面得是一个从句。很明显，这个不是。

C.it
it在这里代替宾语从句to help those people in need
由于feel的实际宾语to help those people in need太长,为了避免头重脚轻,英语中常将其放在句尾,而用it 引导并放在前边,使句子平衡,符合英语习惯.
此句中,it作为形式宾语指代to help those people in need.
Our duty在句中充当宾语补足语
原句相当于We all feel to help those people in need our duty.（显得头重脚轻,不宜）
另举一例：He feels it his honour to serve the people.(他以为人民服务为荣.)

选C
BE动词可以省略。完整的应是：WE ALL FEEL THAT IT IS OUR DUTY TO HELP THOSE PEOPLE IN NEED。