I think there are two dimensions to this.
The first is: what makes a real-world person irredeemable? A fictional character with the same traits will then, presumably, also be irredeemable. I think the already-provided answers by Galastel, Amadeus and Francine DeGrood Taylor do a great job of discussing this idea.
The second is: how does a character function, within the structure of a story, so that, in terms of storytelling mechanics, the character is irredeemable?
At the most basic level, redemption (after extensive evil) requires a deep, challenging character arc. If a character cannot be convincingly given such an arc, the character cannot, mechanically, be redeemed within the story. And that character will also feel irredeemable to the reader, because the reader cannot imagine a situation in which the character is (plausibly) redeemed.
So, for example, a two-dimensional/cardboard character will be irredeemable because redemption has, effectively, not been foreshadowed--we would need a more rounded character to believe that this is a real person, with flaws and failures and blind spots, who could, therefore, come to understand and regret her failures and, thereafter, change.
Also, a character we don't invest in, don't feel any empathy towards, will be (mechanically) irredeemable, because a redemption character arc takes us on an emotional journey that is impossible without us being significantly invested.
A different way to state this. Consider the values explored by the story. The first way to consider the question is about the character's values--the evil character's values are so perverse and despicable that, once they have been translated into actions, we will never forgive the character. The second way is to focus on our values: the irredeemable character cannot guide us on a journey towards a deeper understanding of our own values, whereas a character may be exceptionally evil and yet lead us on such a journey, and thereby, within the story, be redeemed (to some extent) in our mind*. A good example of this is American History X: we will never forgive what the protagonist did as a neonazi, but we are confronted by his humanity in a heart-wrenching way that challenges our way of categorising people.
Moving on to the Harry Potter examples. I think Umbridge is, simply, an excruciating read. All I want is for her to be the hell out of the story, to stop tormenting Harry and everybody else. Whereas Voldemort is, in the storytelling sense, awesome**: powerful, enigmatic and "fallen"--I want to know more about him, and I enjoy myself imagining his upcoming epic defeat, or, who knows, maybe a more complex resolution to the story (although it turns out Voldemort gets nothing like a redemption arc). Umbridge is, thus, further along the scale towards irredeemable because I, the reader, have no interest in seeing that character developed, I just want her defeated, whereas I do want Voldemort developed as a character.
* perhaps I should drop a reference to Aristotelian "katharsis" here--but my classical education is too many decades in the past for me to do so confidently...
** though, personally, I thought he kinda failed as a character late in the series (but maybe this was intentional: when we finally meet him, it turns out he isn't actually that capable at things other than evil magic)