I apologize if this question is not appropriate because I understand and I respect that the ordinances performed within the walls of LDS temples are sacred and holy and are not talked about. However, I know they are not secret (as there are open houses), so I'm not sure if asking about incense burning qualifies as an ordinance or not. If it is, please feel free to remain mum about it, but if it's permissible to talk about -- can an LDS member tell me if incense burning occurs in modern LDS temples akin to the Jerusalem Temple? Why or why not?
No, incense is not burned in LDS temples.
In the temple at Jerusalem, and in some traditional Christian liturgy (such as the Catholic mass), burning incense is used as a symbol for prayer, which ascends to heaven and is pleasing to God. It is also a symbol of the Savior Jesus Christ.
As part of the LDS temple ceremony, participants gather around a table representing an altar to pray. A temple worker kneels to lead the prayer, just as the priest offered prayer in the temple at Jerusalem. Prayer is offered on behalf of those attending, as well as friends and family members with specific needs.
The symbolism and ritual used to teach in temples shares ancient roots with Jewish and Catholic ritual, but has been adapted to meet the needs of people today.
My best response is that the burning of incense was used with burnt sacrifices which were symbolic of the forthcoming sacrifice of Christ. When Christ came, he "fulfilled the law" and live sacrifices were no longer needed.
The LDS faith recognizes Christ as the Savior and having fulfilled that law and thus no longer needing sacrifice (burnt offerings, and incense). Instead, God requires a broken heart and a contrite spirit as our sacrifice, meaning we appreciate the value of the Atonement such that we apply it in our own life through repentance, turning to Him, accepting the gift of the Atonement, keeping the commandments. We partake of the sacrament each week in remembrance of Him. Christ said "in me the Law of Moses has been fulfilled and a new law I give unto thee to love one another...." No longer was the law an eye for an eye but forgiveness (7 x 70) because he had fulfilled law and we can now trust in it. I hope that helps. :)