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He had run barely a dozen steps when he reached them: Dudley was curled up on the ground, his arms clamped over his face. A second Dementor was crouching low over him, gripping his wrists in its slimy hands, prising them slowly, almost lovingly apart, lowering its hooded head towards Dudley's face as though about to kiss him....

[Excerpt from Harry Potter and the Order of the Phoenix by J.K.Rowling]

As I understand, 'apart' means 'separated', and 'lovingly' means "with fondness; with love". It seems strange when they put together "lovingly apart" in this context. I don't know what it's supposed to mean. How should we understand it?

Here, "lovingly" is not modifying "apart". Instead it is modifying "prising". "Prising" is not a word I'm familiar with, but according to Dictionary.com when used as a verb it is a form of "pry". Additionally, the "them" in the phrase refers to "his wrists" from earlier in the sentence.

Thus, I will start with this phrase:

Prising his wrists

Alone this would be a pretty weird thing to say, but there are some additional adverbs to add clarity! The phrase as a whole follows the form "[Verb]-ing [noun phrase] [adverb(s)]", which I can't really put a name to but is fairly common. Lets add the most useful adverb:

Prising his wrists apart

"Prising apart" describes a clear and distinct concept from just "prising", so it is very important for actually understanding the sentence. However, the other adverbs don't modify "apart" they modify "prising" and thus can be considered independently of "apart":

Prising his wrists slowly

You don't seem to have any difficulty with this part, there's no contradiction between "prising" and "slowly". However, the other adverb can also be considered independently:

Prising his wrists lovingly

This phrase does end up being weird, "prying" and "lovingly" seem like a weird pairing, but grammatically it's fine. "Lovingly" also has a modifier that helps:

Prising his wrists almost lovingly

With this, the prising is explicitly not loving, but instead is similar to being loving. If "slowly" is included as well then there's even more detail:

Prising his wrists slowly, almost lovingly

Pairing two modifiers like "X, almost Y" in this way carries an implication that the two are related. Specifically it would most closely be interpreted as "done so slowly that it seemed to be done lovingly".

The way the words connect to each other can be made a bit clearer by changing the word order. This phrase has the same meaning as the original:

Slowly, almost lovingly prising his wrists apart

TL;DR: "Lovingly" modifies "prising", not "apart".

The Dementor is moving Dudley's arms apart in order to take Dudley's soul. The word "lovingly" is used to make the scene more frightful to the reader. Note the word "almost" - this means "had it been in another situation, this movement would have looked very loving and gentle".

There is a contrast between the horrible situation and the gentle prizing apart of Dudley's arms.

There is a horrible scene (a longer version) in Saving Private Ryan in which a soldier almost lovingly and slowly puts a knife in another soldier during hand-to-hand combat.

The indicated phrase means nothing in this context, because that's not really the phrase.

What you should be looking at is slowly, almost lovingly. It's saying that an action (the prying) is happening so slowly that it almost seems loving, and/or that the action is being done with a degree of delicateness (or apparent delicateness) similar to that of a loving touch.

The sentence could have been written with an additional comma after "lovingly", to make it clearer that this is the intended structure of the sentence; but some editors might prefer to avoid that.

As others have noted, "almost lovingly" is a parenthetical phrase. This sentence is poetic, but horrible (and an example of why I don't like my native language much). To be unambiguous, the sentence should read:

A second Dementor was: crouching low over him; gripping his wrists in its slimy hands; prising them slowly, almost lovingly, apart; lowering its hooded head towards Dudley's face as though about to kiss him…

With the added punctuation marks, you can clearly see that the sentence is a list of the actions that the Dementor is performing, and that "almost lovingly" is a parenthetical phrase. But now it seems stiff and formal; the punctuation marks interrupt the flow of the sentence because they not only represent boundaries between grammatical structures but also pauses in the sentence when read aloud.

This is an example of a writer forgoing precision for effect. It happens loads in English. If you find part of a sentence that doesn't make sense, treat it as you would a garden-path sentence and assume that you've parsed the sentence wrong.