Leviim hold holiness in their essence

במדרש שבט לוי לא הי' בחנוכת הנשיאים וניחם הקב"ה לאהרן במנורה ע"ש.

In the Midrash: the tribe of Levi was not included in the dedication offerings of the Nesi'im (the tribal princes), and HaKadosh Baruch Hu comforted Aharon with the Menorah; see there.

The Sefas Emes opens from the Midrash: Levi was left out of the princes' inauguration of the Mishkan, and Hashem consoled Aharon by giving him the lighting of the Menorah.

והענין שהנשיאים נתפעלו ע"י המשכן כמ"ש ויקריבו נשיאי ישראל כו'.

And the matter is that the Nesi'im were aroused by means of the Mishkan, as it says, "and the princes of Israel brought offerings," etc. (Bamidbar 7:2).

The princes were inspired from outside themselves — the holiness they felt came to them through the newly erected Mishkan.

שהרגישו הארת הקדושה בהקמת המשכן.

For they sensed the illumination of kedushah at the erection of the Mishkan.

The princes perceived the radiance of holiness that the Mishkan brought, and that perception moved them to bring their gifts.

אבל הלוים הם בעצמם בכללי כלי המקדש שהלוים נושאים הארון.

But the Leviim themselves are included among the vessels of the Mikdash, for the Leviim carry the Aron (Ark).

The Leviim did not receive holiness from the Mishkan from without; they were themselves part of its sacred apparatus, bearing the Ark.

והכהנים בעצמם נקראו כלי השרת ולכן לא היו בחנוכת הנשיאים שענין החנוכה להיות נשאר בהם הארת המשכן לעולם.

And the Kohanim themselves are called the kelei hashareis (vessels of service); therefore they were not part of the Nesi'im's dedication, for the purpose of the dedication was that the illumination of the Mishkan should remain within them forever.

The Kohanim are themselves "service-vessels," so they had no need of the dedication, whose whole point was to make the Mishkan's light permanent in those who, unlike the Leviim and Kohanim, lacked it intrinsically.

והלוים יש בעצם נפשותם זה הדביקות.

And the Leviim have this deveikus (cleaving to Hashem) in the very essence of their souls.

For the Leviim, attachment to Hashem is inborn — woven into the essence of their souls rather than acquired through the Mishkan.

וכעין זה איתא במד' במדבר שבפקודת בני ישראל נזכר המשכן שבנ"י יצאו ממצרים בזכות המשכן.

And something like this is brought in the Midrash on Bamidbar: that in the counting of Bnei Yisrael the Mishkan is mentioned, for Bnei Yisrael went out of Mitzrayim in the merit of the Mishkan.

The same pattern appears in the census: Israel as a whole was redeemed from Egypt through the merit of the future Mishkan — an external source of holiness.

והלוים יצאו בזכות עצמם.

But the Leviim went out in their own merit.

The Leviim, by contrast, were redeemed through their own inherent merit, needing no external cause.

כי תכלית יציאת מצרים להיות לכם לאלקים.

For the purpose of the Exodus from Mitzrayim was "to be a God unto you" (Vayikra 22:33).

The goal of Yetzias Mitzrayim was that Hashem should become Israel's God — to receive a power of Divinity.

ובנ"י קבלו כח אלקות ע"י המשכן ולוים בכח עצמם:

And Bnei Yisrael received this power of Elokus (Divinity) by means of the Mishkan, while the Leviim received it by their own power.

The nation as a whole absorbed the indwelling of Divinity through the vehicle of the Mishkan; the Leviim carried that Divine power within themselves from the start.

Summary: Levi was absent from the princes' dedication because the Nesi'im needed the Mishkan to awaken holiness in them from without, whereas the Leviim and Kohanim are themselves sacred vessels of the Mikdash, with deveikus rooted in the very essence of their souls. The same contrast appears in the Exodus: Bnei Yisrael as a whole were redeemed and received Divinity through the merit of the Mishkan, while the Leviim went out and possessed that Divine power by their own intrinsic merit — and so Aharon was comforted with the Menorah, a service flowing from within.