没有人不喜欢自己的。 what does 的 imply here?
@Stan's reference to NS.X's answer gives a good explanation. 是...的 construction doesn't necessarily imply past action. It's just for emphesis. It's easy to give some examples for this usage.


  I will never buy iPhone. I only use Android phone. 我是绝对不会买 iPhone 的。我只用 Android 手机。


NS.X's answer said that the 是 … 的 construction emphasises time, manner or place. In fact, it can also be used to emphasise the action(verb) itself. In 我是绝对不会买 iPhone 的, 的 emphesizes 不会买(won't buy). 是 can be left out here. (You can also say that the 的 emphesizes the whole sentence.)

With regard to没有人不喜欢自己的, the 的 is used here to emphasize on 不喜欢(doesn't like). Its full version is 没有人是不喜欢自己的. 是 is left out here.

Look at the fourth meaning item of 的 from Contemporary Chinese Dictionary, 6th Edition. 

Figure 1. Contemporary Chinese Dictionary, 6th Edition, page 272


④用在陈述句的末尾，表示肯定的语气：这件事儿我知道~。
It can be translated as "Used at the end of a declarative sentence, indicating an affirmative tone: I do know this thing."