According to the Mishnayos both the mitzvah of ראשית הגז and the mitzvah of מתנות כהונה apply without a Beis HaMikdash and even outside of Eretz Yisroel. However, Rav Nachman 136b made a statement that the minhag is to follow רב אלעאי by ראשית הגז and that it should only apply in E.Y. Rashi comments - והוא הדין במתנות, meaning that there is no rationale to differentiate between ראשית הגז and the matanos of זרוע לחיים וקיבה. Since the minhag has become to follow רב אלעאי by ראשית הגז, we also follow him for matanos kehuna that they don't apply outside of E.Y. The Shulchan Aruch both in the halachos of Matanos Kehuna (y.d. 61:21) writes that they apply even outside of E.Y. and then cites a יש מי שאומר שאינן נוהגות בחוצה לארץ. The Shulchan Aruch in the context of matanos kehuna writes וכן נהגו on the lenient opinion even though the main opinion seems to obligate matanos even outside of E.Y. Interestingly, the Shulchan Aruch in the context of reishis ha'geiz doesn't even quotes the stricter opinion - he writes that it only applies in E.Y. to which the Rama adds that some says it applies even on a Torah level in chutz la'aretz but we aren't noheig to be machmir. The Shulchan Aruch is more confident about exempting from reishis ha'geiz in chutz la'aretz since that is explicit in Rav Nachman's statement. Although there is no rationale to differentiate between reishis ha'geiz and matanos, the gr"a (61:20) explains that the leniency is based on a minhag which may have only been to be lenient about reishis ha'geiz and not about matanos kehuna. The Beis Hillel (333) struggles with trying to understand why we are even lenient in E.Y. nowadays. Although the Pischei Teshuva (61:8) cites the chasam sofer who would routinely shecht an animal before yom tov and fulfill the mitzvah of matanos kehuna and reishis ha'geiz, the minhag is not to be makpid about fulfilling these mitzvos. The Beis Ephraim (on shulchan aruch 61) quotes the kerisi u'pleisi who wonders why we don't find people going out of their way to fulfill this mitzvah. He quotes the Pri Chadash who argues on the Beis Hillel's (who wrote that he was told that even in E.Y. people don't fulfill the mitzvah of matanos kehuna), and says that they didn't realize what they saw because it is מעשים בכל יום that matnos kehuna are given in E.Y. from the animals that are being shechted. The Beis Ephraim also quotes that the kreisi writes that his son in law insisted on giving matanos kehuna at the pidyon ha'ben of his son. Since he was being machzik the kohein as a kohein for the purpose of discharging his pidyon ha'ben obligation, he certainly can rely on his yichus to give him matnos kehuna.