The attribution of frankincense to the Sun is standard, and it is worth being exact about where it comes from, because the usual citation is wrong.
It is not Culpeper. The Complete Herbal of 1653 is an English book about English plants: Culpeper wrote for people who could walk out and gather what he described, and a resin imported from southern Arabia is outside that scope. Looking for a Culpeper rulership of frankincense and finding one quoted is a good sign that the quoter has not opened the book.
The solar attribution belongs instead to the older magical tradition. Agrippa lists frankincense among the suffumigations of the Sun in the Three Books of Occult Philosophy of 1533, in the section setting out which substances are burned to draw down which planetary influence. That is a Renaissance systematisation of material Agrippa had from Arabic and Hermetic sources, and it is the proper reference.
Behind Agrippa there is something older still, though it is not astrological in his sense. Frankincense was burned in temple rites across Egypt, Mesopotamia, Israel and the Greek and Roman world, in cults of deities who were solar and deities who were not. The association with the Sun in a technical planetary scheme is a later tidying of a much broader ritual fact. Both things are true, and they should not be collapsed: the incense is genuinely ancient, and the planetary label on it is sixteenth century.
There is a plain physical logic to the assignment that is worth stating. Frankincense is golden, it is costly, it rises, and it was burned in the temples of gods addressed as light. Any scheme sorting substances by planet was going to put it under the Sun; the surprise would have been anything else. That the answer is over-determined is a reason to hold the attribution loosely.