Please help me understand this use of 去做

但是很多时候，我们不得不去做一些自己不愿意做，甚至是非常不喜欢的工作
However, there are times we have to go do something that we hate...

The 去 go here functions in contrast to "hate" in the second part of the sentence.
It shows how much you hate to do it, but also need to do it. So you "go do it" anyway.

In your example,

我们不得不去做一些自己不愿意做

with 去, it adds a bit emotion here. It sounds like that doing those things is extra miles. Without it, it doesn't have that connotation.

This dictionary definition addresses the use of 去.

（用在另一动词前， 表示要做某事）(preceding another verb, denoting will do):

Please go and ask.

你去问一问。

We'll find a way out ourselves.

我们自己去想办法。

去 means 'go'. It can function as a verb particle that denotes the initiation of a verb. Similar to 'go' in 'go kill people' in English

More examples:

我當教師 = I being a teacher

我去當教師 = I go and (begin to) be a teacher

因窮而犯法 = break the law because you are poor

因窮而去犯法 = go and (start to) break the law because you are poor

很多时候，我们不得不做一些自己不愿意做 = We often do something we don't want to do

很多时候，我们不得不(去)做一些自己不愿意做 = We often (go/ start) doing something we don't want to do

A simpler example:

殺人 = kill people

去殺人 = go kill people (initiate the action 'kill')

It's absolutely ok to remove the ‘去’. The meaning won't change at all.

去 may put a little emphysis in the sentence (because it makes the sentence longer), but it's largely redundant anyway.