What is the problem, then, if it is purely kosher Lurianic kabbalah?
1. The answer is that it could have been Sabbatean.
To elaborate, I will first go off on a tangent.
If I were malicious enough, I could convince plenty of frum Jews to (unwittingly) offer their child to Molech, in violation of the Biblical command.
How so? I would tell them that it was an old chassidic/kabbalistic custom, encouraged by (insert obscure chassidic or kabbalistic master here), and the practice was to line up two rows of lit candles. The meaning is "deep," but partly it is a reenactment of bris bein habesarim, but also because the right row of candles represents chessed while the left row of candles represents gevurah. And one should pass the infant between the two rows of candles, just in the middle, to allow him/her to achieve a proper balance of chesed and gevurah. (Of course, we are not talking about burning the infant, just passing him between the two rows of fire.) I would also give them a "kabbalistic" prayer to recite while performing the ceremony, in arcane Aramaic and even then speaking in code where possible, perhaps even using "Malcam" which they may interpret as "their king."
This is because there is an opinion that the Molech offering was not actually human sacrifice, but rather passing the child between two fires.
I wouldn't actually do this, of course, but if I did, and with the appropriate attachment of segulah and mystical explanations, I could probably get quite a few people to do this.
And this obscure custom would spread by email, even to people who had not practiced it before or did not come from kabbalistic / chassidic roots, just like the practice of saying parshat HaMan specifically on Tuesday of the week of Beshalach.
People are careful about the food and drink they put in their mouths but are not so discriminating in the rituals they practice and the tefillot they say.
Most of the people practicing Tu BeShvat seders today, who do not have a family tradition from years past, have no idea of the Sabbatean connection -- of the text coming from Nathan of Gaza's anthology, and of Rav Yaakov Emden's condemnation. They do not know that there was controversy, but it is OK because it turns out it is actually entirely Lurianic kabbalah.
In fact, they probably have no idea of the meaning of the words in the seder, for even if they know the straightforward translation of the words, they do not know the import. How many people who say kegavna instead of bameh madlikin know the straightforward translation of the words, let alone the meaning of the statement, even if they read it in English? You need to be well versed in kabbalah to understand its import, and the same is true to really understand the Tu BeShvat seder. (You can see an abridged English text here, and the full translation [though some pages are not in the preview] in Trees, Earth and Torah -- A Tu B'Shvat Anthology starting on page 135.)
Aside from this, given that it is Lurianic kabbalah. Is it Avodah Zarah? Most people are not kabbalists, and don't say kabbalistic tefillot, so they do not come across this issue. But the Tu B'Shvat seder is concerned with the sefirot, which Shadal in his Vikuach al Chochmat haKabbalah charges is avodah zarah. The typical person who newly adopts this seder wants to do a new cool Jewish thing, but is not educated enough in kabbalah to make such a determination. (Of course, one can rely on greats who do not hold it is avodah zarah.)
That is the first issue.
2. The second issue is the general tendency to adopt new practices, be they hafrashat challah segulas, saying parshas Hamon during Tuesday of Beshalach, or Tu BeShvat Seders. This is somewhat related to the lack of discrimination mentioned above. The rituals we do have do not have enough significance for us, and so we seek to innovate, or adopt, new rituals.
Drinking four cups of wine and saying things over them (and over fruits), as part of a seder, and as a new instituted practice on a specific day of the Jewish calendar -- why is this not a violation of bal Tosif? There were issues concerning Purim (see Yerushalmi Megillah here, starting at the bottom), so why is there no problem making Tu B'Shevat into a chag, with its own rituals and tefillot?
The answer may be that there is no problem on the level of halacha, but on a conceptual level, couldn't this adoption of new ritual and prayer on specific days (Tu BeShvat seders and reading parshat haMan specifically on Tuesday of parshas Beshalach, or tens of other examples) reflect an attitude of Bal Tosif.
In terms of parshat Haman, Artscroll is getting in on the action this year. They give a text and translation of what to say this Tuesday, Jan 15, parshat B'shalach, in a downloadable PDF. Get it here. They write:
The Torah reading of this coming Shabbos -- Parashas Beshallach -- includes the chapter telling how the Jewish People in the Wilderness received manna, the Heavenly food that nourished them for forty years. Many people recite the chapter daily, as a special prayer for parnassah. There is also a widespread custom to recite this chapter on the Tuesday of the week of Beshallach, which this year is January 15.
As a public service, we offer the text and interlinear translation of the chapter, from the Schottenstein Edition Interlinear Siddur. Please feel free to download it.
So the practice now has the Artscroll seal, as a widespread custom. Perhaps it really is a widespread custom, but perhaps it is only widespread in recent years as a result of email campaigns of people forwarding it.
Don't take this as a final say not to say it, or to say it. These are just my thoughts on the matter.