How many different perspectives do you have or want to have? Would it make sense for each of them to be first-person? Then close third would probably work just fine, too.
I stay away from omniscient, but it makes most sense when you have a big gallery of characters and want to reserve the possibility of dipping briefly into a POV we never see again after that. (I don't know Harry Potter that well. But once I read a book that ended with a single chapter from the POV of a crow. And not a magic talking crow, either. The author just dipped into that POV so he could show the crow finding something the human characters had lost, and it felt pretty silly IIRC.
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A good example of close third with multiple POVs is We All Looked Up (I just started reading it). The characters get alternating chapters, and each chapter is firmly in the character's voice, but it's still third person. It never feels like there's some overarching narrator, which is what you get with omni. It can be hard to write voicey third, but it's a great skill to develop, and I admire writers who can do it well.
That book is in past tense. Present tense seems less common in close third books, though I've seen it (Pure by Julianna Baggott comes to mind).
Me, I've been writing first person, present tense with multiple POVs (usually just two), and I really enjoy it. It's almost like a dramatic monologue; it's a great way to channel a character's voice.
Of course I worry about whether the POVs sound too alike; that's always a concern. I think the method works best when the POV characters have radically different voices and perspectives on the world, and are often in conflict within the story. Gone Girl is a good example: it would be hard to mix up Nick and Amy. It's a really fun way to deal with antagonistic characters who both think they're the hero of the story.