How long can you ignore Paimon?

Depends on your definition of "ignoring her" and still being able to play the game.

If you try to ignore Paimon right from the start then your best bet is to never approach her as she is waiting near the lake where you find the first statue of the Seven.

This... causes quite a bit of issues.

• since you never interacted with the statue, you will never get Anemo. Meaning you won't have any elemental skill / burst on the main character. You probably could use wishes to get other characters, but .... we will talk about that in a bit.
• the main quest won't progress, so you never meet Amber. Meaning no glider.
• other world quest elements seems to be stuck also. For example some claim that the shop in Mondstad are stuck in a "we are about to open" limbo forever.
• also, most of the other quests either have a minimum prerequisite advancement in the Archeon main story OR an AR requirement. So, you can't play most of them.
• you could think to outsmart the game by exploring and collecting chests to raise your AR.... but apparently the map is also stuck until you advance the story. This means that the red barrier is at full power, and you can't even explore the full Mondstadt region.
• your low AR will keep you out of most events and free giveouts, so your free primogems income won't be that much. You probably could still buy stuff with real money.. but it is really worth to spend on a butchered game?

Also, even if you somehow managed to find a way to workaround all of this (bugs? I doubt there is a way to go past the map borders but again the Dokodemo Door in Liyue apparently survived for a few patches...) I once again wonder what "ignore Paimon" means. I suspect that she would still talk to you if you crossed some trigger areas where she comments on landmarks and/or nearby zones, she would still comment on random event quests (like Pallad, assuming those will trigger in the first place since you never really started the main quest line), and she may even automatically start a dialogue upon entering the game when some events are running if your AR was high enough. Maybe you could skip those by trying to keep you AR low, but in that case probably you won't be able to really "play" the game.

Also worth noticing that on a pure "Lore" level, Paimon seems to be fully aware of the actual player playing the game since she directly speaks at you in the setting menu (Which not so randomly is called "Paimon Menu"). So, you could say that you are interacting with her whenever you start the game in the first place.

Anyway, if you are curious, this Youtube serie tried to do that.

No Element Traveler - Playing Genshin Impact without Elements

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UPDATE:
Other answers pointed out that you will get to a breaking point when you reach the Ascension Quest that has you battle an Electro Hypostasis since you have no way to destroy the cores it uses to regenerate itself. You don't have access to wishes since you never advanced the main quest (requires beating the prologue), you don't have access to the free Barbara for the same reason and even the free Xiangling from the Abyss challenge is out of grasp since you can't reach the Abyss in the first place.

The Geo Traveler could abuse out-of-bound exploration to skip most of the Ascension quest as does Venti and few other characters but you can't access them either. No-Element traveler doesn't even have access to gliding so hopes to skip the domain are very slow.

Your best bet would be to abuse some critical level bug that let's you teleport around the map like the aforementioned "Dokodemo Door" did (a bug that once existed in 1.2. Never seen it firsthand but it was confirmed from multiple sources so it was probably true?). That said the Door seemed to be a fail safe protection that was triggered when the game decided that the player had invalid coordinates, probably a piece of a debug feature. Lately, I firsthand saw the game just straight killing you in the Archipelago while going in the mist at the north point of the map and the game could not force you back to the respaw point at the island south of the mist. The game tried to load your location a few times... and then put you back at the Resurrection Gate. I think this is part of a patch to the Docodemo glitch.

At this point, I think this is a lost battle but your only hope to "progress" in this butchered experience is to find a way to glitch out the Ascension domain, hoping that it doesn't get you banned in the process. But as they said in Morrowind....

"With your stubborn ignoring of your best companion, the thread of prophecy is severed. Speak to Paimon to restore the weave of fate, or persist in the doomed world you have created."

Well, ignoring Paimon means that you will need to play with bugs, otherwise you don't have a lot of things to do.

You could ignore Paimon, and go straight to Mondstadt, but without reacting with Paimon, you could said that the time stops running, you will never meet with Amber and other characters.

Let's call this playstyle "no paimon" for short in this answer.

In "no paimon", you can't play with other people, other people can't join your world, and you can't join their world too. You can't glide using wings, no wings because you never meet Amber(okay, "no paimon" can use the boat in Golden Apple Archipelago and Inazuma to glide a little bit when you get out of the boat.) You can't open the event page, except in the Energy Amplifier Initiation event, it opens the event page after chatting with the NPC for once, and it bring flame to "no paimon" playstyle like what Prometheus does（Well， that requires some usage of bug to get to Spiral Abyss). Also, you can't open mails, which means "no paimon" can't get the free 5 star character.

Your main character can't use any element in this playstyle, and you can't get any other character except Xiangling by using bug to get to Spiral Abyss first(in earlier version, now the bug is fixed.), and complete 3-3, then, you need to have a teleport waypoint outside of the red circle(by using bug in earlier version, fixed too.), and that waypoint has to be able to get to the NPC of Energy Amplifier Initiation event with the paimon by your side saying that "How about we explore the area ahead of us later" and control your character moving to inside the red circle.

Primogem almost worth nothing in "no paimon", you can't wish at all. The only usage of the primogem is that you could use primogem to buy some level of battle pass.(You can't buy Genesis Crystal, but you could buy battle pass.) Maybe you will say that primogem can buy Original Resin, but Original Resin worth nothing before you have Xiangling, Xiangling effect the "no paimon" playstyle greatly.

Before you get Xiangling, “no paimon" can't get over 25 in Adventure level, means no 5 star Artifact, even 4 star Artifact are very rare too. Also, character level is lock on 50. It is because the Electro Hypostasis in the world level upgrade mission, you don't have any element to destroy Electro prisms to kill the Electro Hypostasis. But with Xiangling, Electro Hypostasis is killable, and means that the adventure level can get over 25 now.

After all, "no paimon" is not that playable right now, you can't get Xiangling now, so no way to bypass Adventure level 25 now, and no waypoints outside the red circle too, those all need bugs, and the bugs is all fixed right now. And you could see that "no paimon" is a playstyle that is focused on fighting with programmer(XD).