Any time it furthers the purpose of your writing.
When you write, you're doing it for something - maybe you've got an idea burning in your mind and you feel compelled to put it into writing, maybe you're trying to make the reader feel something or think about a topic, maybe you just enjoy creating a story.
So identify that purpose, and then decide whether or not "a thoroughly despicable character with no redeemable qualities" serves it.
In general, characters like that play a useful role in:
1. Stories where the despicable character serves as a force or influence on the characters your story actually focuses on.
Example: <PERSON>: <PERSON> does much that's despicable, and the story works just as well whether or not the reader feels he has redeemable qualities. And even though his backstory is explored and there is a suggestion that to some extent he may have been the product of his circumstances, and he is clearly portrayed as a gifted, resourceful, and competent wizard, and the books strongly suggest that he could choose to stop generally doing evil things, he chooses to keep on his path to the very end, so many readers can walk away still not liking <PERSON>.
1. Stories where the intent is to cause emotions like fear or dread, or to explore the space of how we normal humans cope with and interact with the "monster" humans. Sometimes you can't invoke those feelings if the "monster" is relatable or likable or even understandable.
Example: If I kidnap you, and tell you I'm just holding you for ransom and explain that it's because Nana's in the hospital with ovarian cancer and I already lost my mom when I was young to the same thing and I just can't bear to go through that again - that invokes a completely different feeling than if I tell you that I'm just a sadist, sucks to be you, oh check out this drill, it's about to go in your knee-cap. And quite the emotional roller coaster if I tell you the former but then act like the latter.
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Stories where the focus isn't the exploration of the despicable character, but their effects on the world and on people.
Example: If you're exploring the damage of and recovery from child sexual abuse, your child-raping character doesn't need to (and indeed probably ought not) be explored for redeeming qualities. (There can be stories that meaningfully explore both, but they're extremely hard to pull off, will enrage countless people for merely daring to exist, and the purpose for such stories is more nuanced and different than the purpose given in this example.)
Regarding your advisor's advice:
Each character deserves the chance to be liked by the reader
To me this doesn't mean they can't be despicable, or that they need to have redeemable qualities.
It just means you have to give the reader enough insight into the character to allow them to make the judgement call about the character for themselves.
A "complete monster" will fail their "audition" to all or most readers: what you're not supposed to do is skip the audition entirely - at least, if you want the character to be a real character, a fleshed out being.
But a character doesn't have to be a character.
You can of course also have characters who are just anthropomorphized forces or play a role more as symbols rather than as people. They won't "make sense" to the same careful inspection that a fully fleshed out character does, but they're arguably not supposed to.
The <PERSON> is generally held to be an effective villain in <PERSON> stories in part because he mostly plays the role of a philosophical rejection of much that the protagonist stands for, and generally is a chaotic unpredictable opposing force. He invokes a wide range of emotions in the audience, but mostly while staying in the rather despicable territory in most of his appearances.
Now that said, note that "despicable" and "redeemable" are subjective.
For every well-developed character, no matter how despicable and irredeemable to the vast majority of humanity, you will eventually find a mind which will not agree with your assessment that your character ought not be liked.
Ask yourself why you don't want your readers to like the character. Really take a look at why you don't want the character to be liked by anyone: This will probably help put into focus why you want to write the given story a given way, besides being just good self-awareness mental hygiene.
As a parting exercise:
1. Read and consider the short story The Things.
2. The consider and compare the story of The Thing (1982).
Think about how neither story works quite the same if you were to bring the characterization from one into the other. 425