It means something can't be taken. Let's say you give someone something and they think it's too valuable to take. Or someone tells you from the outside that you can't have it.

can't keep havin' it?

There are two cases： 1. Someone want to give you something，and others may say“这个东西不能要”，that's mean you cannot take it whether you want it or not，maybe because “这个东西”is very expensive or not so good. 2. If you say“这个东西不能要了”，that usually means it is broken or rotten，and it is useless so you don't need to keep it any more.

ไม่ควรมี ไม่ควรเอา 我感觉的是不应该把它过来， 是吧?

在你给的语境中，“不能要” 的意思是 “你不应该持有它” 我们一般在买一样物品或者接收别人的礼物的时候会这么说。 买一样物品的时候说 “这个东西不能要 “ 一般是因为这个product的品质不够好或者 价格不符合预期。 接受别人礼物时说：这个东西，不能要” 一般是由于礼节上的习惯（protocol）或者是由于其他的factors e.g。可能是风俗上或者精神上或者宗教上的