Terumah is a tithe that is given to kohanim. I have been told that kohanim are not allowed to eat terumah nowadays. Is this true? If so, why not?
The kohen needs to be in a state of taharah (ritual purity) to eat terumah. Since without having a parah adumah (red heifer needed to eliminate the effects of the impurity of connecting with a human carcass) it is not possible to be in a state of taharah nowadays, the terumah is not eaten.
Terumah [must be given] to a priest whether it is in a state of ritual purity or not. Even if all the grain or the wine became impure before [terumah] was separated, he is obligated to separate the terumah that is impure and give it to a priest, as [Numbers 18:8] states: "And behold I have given you the watch over My terumah,"54 i.e., whether it is pure or impure. The pure [terumah] may be eaten by the priests and they can benefit from the impure [terumah] by burning it.55 If it is oil, it can be kindled [as fuel for a lamp]. If it is grain or the like, it can be used as fuel for an oven.56
54. In the verse, the noun terumah uses a plural form alluding to two types of terumah: pure terumah and impure terumah. See also Hilchot Ma'aser 6:2.
55. It is forbidden to partake of it.
56. It may not, however, be given to animals as animal fodder. See Sefer HaMitzvot (positive commandment 90) which states that it is a mitzvah to burn impure terumah. Impure terumah from fruits that will not be useful as fuel must be buried.
See this source from Orthodox Union, in the section Explanation of the Procedure in the first item.