What does it mean to offer incense burners in Tantric Buddhism?

Although Tibetan Buddhism (i.e. "Tantric Buddhism" as you put it) is different from our Chinese Buddhism, the essence is the same, so the role of the incense burner, are to incense ah.

-By the way, the significance of the popularization of incense:

The incense does not lie in the number of times, and does not emphasize the time, generally is the morning and evening each time, the rest of the time with their own mind and convenience. The key is to understand the true meaning of incense:

First, the expression of piety and respect for the Three Jewels, as a demonstration of the attraction of living beings.

Secondly, it is to convey the message to the Dharma Realm in the void, and to feel the blessing of the Three Jewels in the ten directions.

Thirdly, it means that burning oneself and spreading the fragrance of the Ten Directions reminds the disciples of the Buddha to be selfless and devoted.

Fourth, it means that the Buddhist precepts of the true incense, with a vow of silence, "diligently cultivate the precepts, determination, wisdom, quench greed, anger, dementia," meaning that the Buddha is not addicted to the world's big incense incense, but like the Buddha's disciples of the precepts, determination of the true incense.

Additionally, it is important to note that:

When offering incense to the Buddha, you should have a pure heart, and if you can keep it clean, you will be blessed with boundless blessings. If you want to make a wish, you should give up selfishness and self-interest, and make a big wish to benefit the society and all living beings, and then you will get a lot of merits. It is also said in the Buddhist scriptures that "a single bow to the Buddha will extinguish the sins of the river sand; a single recitation of the Buddha's name will increase the number of blessings".

Three sticks of incense at a time is appropriate. This represents the "precepts, determination, wisdom" three non-leakage learning; also represents the offerings to the Buddha, the Dharma, the Sangha, the everlasting Three Jewels. This is the most complete and civilized incense offering. It is not the amount of incense, but the sincerity of the heart that counts, as the saying goes, "burn three sticks of civilized incense, and pay respect to a piece of sincere heart".

Buddhist scriptures say, "Among all offerings, the Dharma is the first." The so-called Dharma offerings are the teachings of the Buddha's scriptures, which should be read and understood in a rational and legal manner, and should be used to transform others and to speak for others. That is, in daily life and in dealing with others, "Do not do all kinds of evil, and practice all kinds of good deeds."