What is the signification of burning incense during mass and other liturgical functions?
One commentary on this survives from a time when the Roman Catholic Church was united with the eastern Sees. It is from a work in Greek by Germanus I, Patriarch of Constantinople from 715 to 730, entitled Ecclesiastical History and Mystical Contemplation, available in a dual Greek-English format from St. Vladimir Seminary Press (Germanus is regarded as a Saint within both the Eastern Orthodox and Roman Catholic Churches).
Although he is explaining the eastern, Byzantine liturgical rite and not the western, I think that his explanation would also be accepted within the context of the Roman Catholic Mass:
The censer demonstrates the humanity of Christ, and the fire His divinity. The sweet-smelling smoke reveals the fragrance of the Holy Spirit which precedes. For the censer denotes sweet joy.
Again, the interior of the censer is understood as the [sanctified] womb of the [holy] virgin [and Theotokos] who bore the divine coal, Christ, in whom the whole fulness of deity dwells bodily (Col. 2:9). All together, therefore, give forth the sweet-smelling fragrance. Or again, the interior of the censer points to the font of holy baptism, taking into itself the coal of divine fire, the sweetness of the operation of the Holy Spirit, which is the adoption of divine grace through faith, and exuding a good odor.
Incense has been used in the Church for centuries.
We do not know exactly when the use of incense was introduced into our Mass or other liturgical rites. At the time of the early Church, the Jews continued to use incense in their own Temple rituals, so it would be safe to conclude that the Christians would have adapted its usage for their own rituals. - WHY IS INCENSE USED DURING MASS?
As for what is the signification of burning incense, this pleasant sweet aroma of the incense signifies our prayers being taken up into heaven and into the presence of God.
Incense is a very ancient symbol of prayer and offering. Archeological discoveries have suggested that humans may have been using incense for religious purposes as far back as twenty-five thousand years ago. The ancient Hebrew people continuously offered incense to God in the Temple as a symbol of their prayers. In the Psalms, the hymns of ancient Israel, the poet sings, “Let my prayer be set forth in your sight as incense; the lifting up of my hands as the evening sacrifice.” (Psalm 141:2). The incense smoke that silently rises to heaven is a symbol of our prayers carried heavenward to God.
The Book of Revelations enriches this imagery as an angel “who had a golden censor, who came and stood at the altar. A large quantity of incense was given to him to offer with the prayers of all the saints on the golden altar that stood in front of the throne; and so from the angel’s hand the smoke of the incense went up in the presence of God and with it the prayers of the saints.” (Revelations 8:3,4). Using the imagery of the worship of God in Heaven itself, the Church uses incense to worship God here on earth where our prayers rise and are joined with the prayers of the whole company of heaven assembled around God’s throne. - Holy Smoke! Why do we burn incense?