However,it's important that lovers not become too dependent on the other person.

本句，在 that 引导的主语从句中，省略了 should 的缘故：

However,it's important that lovers should not become too dependent on the other person.

这是虚拟语气在名词性从句中的用法。虚拟语气用在主语从句、宾语从句、表语从句和同位语从句中。

在以下结构的主语从句中，谓语动词用“should + 动词原形”，should 可以省略。

It is/was+important that...型：

(necessary,appropriate,proper,right,essential,vital,natural,unbelievable,strange,surprising) that...

    It is/was very important that one（should）master the skills of computer.

    掌握电脑知识是非常重要的。

    It is surprising that you（should）not understand me！

    你竟然不理解我;真是令人惊讶。

    It is necessary that we (should) keep the balance of diet.

    保持饮食均衡是很必要的。

    It is vital to our country that we (should) develop economics.

    对于我们国家来讲，发展经济是至关重要的。

    It is strange that she (should) be so lively.真奇怪，她竟然这么活泼。