Non-kohanim were not allowed to eat the special portions given to the kohanim, including terumah and bikkurim (first fruits).
The reason for this mitzvah is that terumah is the special gift given to the kohanim in exchange for their service in the Temple. They could use it to feed members of their household, for whom they are responsible, but they cannot give it to outsiders. It’s special for kohanim and non-kohanim should not mix in on this special arrangement between God and His chosen priests.
This mitzvah applies to all non-kohanim when the laws of terumah are in effect. It is discussed in the Mishna in the seventh chapter of tractate Terumos and in the first chapter of tractate Challah. It is codified in the Mishneh Torah in the sixth and in the eighth through tenth chapters of Hilchos Terumos. This mitzvah is #133 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos.