“Honey” can mean more than one thing in Hebrew. For sure, it means bee’s honey, but it also includes date honey and other fruit products. All of these are prohibited to be offered on the altar.
The reason for sacrifices is to inspire in a person certain thoughts and feelings. Leaven and honey are representative of certain characteristics that ought not be part of the sacrifice process (refer back to Mitzvah #9 for a partial list of negative traits represented by chometz, leaven).
Here, the Sefer HaChinuch speculates that leaven, which takes a long time to rise, represents lethargy. Its omission reflects the opposite trait, that of zeal. Leaven, which inflates in size, also represents ego and self-importance. Leaving it out represents humility. Similarly, honey and fruit juice, which are sweet, represent desire and gluttony; their omission represents the ability to control one’s physical urges.
One was only prohibited to bring these things as an offering on the altar. If someone used them as kindling for the fire of the altar, that might be weird, but it would not violate this commandment (see Talmud Zevachim 77a).
The Rambam (Maimonides) counted the prohibition against offering honey and leaven as a single mitzvah, a “lav she’bichlalos” (a prohibition that forbids several different things – refer back to the Rambam’s ninth principle for determining the 613 mitzvos). However, the Ramban (Nachmanides) considers them two distinct prohibitions because the Torah specifically excludes leaven elsewhere (in Leviticus 6:10).
This mitzvah applies to male kohanim when the Temple service is in effect. It is discussed in the fifth chapter of the Talmudic tractate of Menachos, in large part on pages 58a-b. It is codified in the Mishneh Torah in the fifth chapter of Hilchos Issurei HaMizbe’ach. It is #98 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.