I like The Ferrett's take on it:

New Relationship Energy always reminds me of the way Cosmic Power is handled in comic books. Everyone wants it. Everyone thinks they can handle it. But once they start fooling around with Phenomenal Cosmic Power, everyone either goes on a rampage or goes completely insane, or both.

NRE is potent stuff, man. It's that intoxicating feeling at the beginning of the relationship where your new lover is so sparkly and neat and everything they say is funny and even their bad habits are cute and OMG I DO THAT TOO! And you fall in love with this wonderful person because everything is a new discovery, and if you're not careful you disappear from sight because sure, you have friends, but are they as cool as Schmoopie over here? I think not.

[...]

Thing is, the long-term stable poly relationships are often much stronger than monogamous relationships - and that's because used properly, NRE can fix problems you didn't know you had. Because in any long-term relationship, you tend to just go numb to the things your partner's bad at providing. Not that you didn't try earlier, but you've come to accept that your lover isn't particularly romantic, or they can't take criticism without flying off the handle, or they're bad about being there for you at the end of a long work day. You tried enough times, and now that's a dead space.

You know what, though? New Lover's good at that. New Lover's reawakening parts of you you never knew you had. New Lover is connecting with you on emotional channels you'd flipped off.

The trick is not to switch all those channels over to New Lover.

If you're an old hand at NRE, that's when you go to your partner, without comparing, and say, "Look. I've been dating Jamie, and she's been really good about giving me lots of cuddles when I'm down. I can get it with her... But I want it from you."

If you're lucky and open in your communications (and careful not to pass judgment), you can make your old partner realize that these are things you really need, and hopefully s/he will try once again to open up a channel you'd closed a long time ago. You don't want to do that with everything, but used properly, NRE can have you recognize what's missing in your old relationships - and then try to make that happen.

...which doesn't negate the new partner. Chances are, if Jamie's all good at cuddling, s/he's going to be better at it than Old Partner simply because it's in her nature. Some people are just more inclined to do certain things. But just because Jessie's a soppy romantic doesn't mean that your old-and-stiff partner can't learn to bring home flowers once in a while. In a way, it means more from your older partner because it's not their nature, and when they do it it's a purer expression of love.

In this way you can come to realize what's critical to your well-being, because a lack is never sharper than when it's being fulfilled elsewhere. The trick is not to see new partner as an escape, but a lesson in "What makes you truly satisfied" that can be put to use elsewhere.

Also, his clarification in the comments:

If you go about it in the sense of "X does Y, you don't," then yeah, you've fucked up. But to say that "X gives me Y, and I've really missed getting that from you" is an entirely different thing.

If you bring it up as "You're inferior to X," then yeah, you're a dope. But saying, "I love you, and it hurts me that this is a lack in our lives, because X can do it, but I'd rather have it from you," is something that, I've found, is often both flattering and revealing.

Of course, every partner reacts differently to things. But trying not to frame it in the context of a new relationship makes it seem much closer to lying to me, because it's blatantly apparent to everyone that it is.

Many polyfolk deal with this sort of thing, much as people in monogamous relationships deal with their partners becoming absorbed by a new interest, being assigned to six-month deployments overseas, driving trucks for weeks at a time across the country, having crushing deadlines at work, or otherwise having things come up in their lives that force their partnership to take second priority for a while.

The common thread in my experience is an acceptance that they are not the absolute top priority in their partner's life (partners' lives) 100% of the time, and that's OK, and the relationship is still positive and valuable.

So, yes, that is something that some poly folk can deal with in a polyamorous context.

Whether it's something you can self-modify out of (second person used advisedly), I don't know.

I'm not such a person, but I've dated poly people who seemed to hyperfocus on their new love interests like that. One in particular stands out as someone who'd become deeply infatuated with the current object of attention, almost to the exclusion of others.

Said person was also very new to introspection, rather comfortably selfish (that's not a "boo!" signal, just a relevant and somewhat abnormal trait -- they didn't have much empathy or concern for the feelings of others if it didn't impact them directly and insofar as they knew it might cause others to feel hurt, didn't want to self-modify), and wasn't very able at the time to understand people feeling hurt as anything other than an attempt to manipulate due to a lengthy abuse history.

I'm sure that there are people closer to "baseline" (whatever the heck that is) who are poly and do this. I do get rather intense NRE, and my feelings for each of my partners are somewhat different, but it still doesn't wipe out the feelings for other people. I think the advice I'd give such a person, if they wanted to change this for the sake of their partners, would be to cultivate a lot of self-control, and maintaining perspective. Your new love may push different buttons than your old love, but what you're experiencing is a neurochemical rush which will not last -- when it passes, you and your existing loves will either be grateful it's over, or picking up the pieces. In short, I treat NRE as something on the order of puberty or psychoactive drugs in terms of its emotional intensity: be aware you're extremely biased in this state.