The Mishna says that one cannot harvest before the Omer (according to some girsa'os it should say before pesach). The Rambam (Hil. Temidin U'musafin records this halacha by saying
אסור לקצור בא"י מין מחמשת מיני תבואה קודם לקצירת העומר שנאמר "ראשית קצירכם" שיהיה תחלה לכל הנקצרים
The Rambam clearly holds that this halacha is limited to harvesting inside Eretz Yisroel. He also clearly writes that the heter to harvest is not dependent on the bringing of the korban o'mer, rather on the harvesting of the o'mer. Although the Kesef Mishna asks that the gemara 70b implies that one cannot harvest before the bringing of the omer, the lechem mishna explains that the Rambam's ruling of it being dependent on the harvesting of the omer is clear from Rashi 70b that harvesting is already permitted on motzei yom tov, immediately after the harvesting of the omer.
However, it is unclear whether this prohibition applies even in E.Y. after the destruction of the beis hamikdash when there is no longer a korban omer. The Sha'ages Aryeh (chadashos 8) explains that the Rambam holds that it applies nowadays as well, since he doesn't limit the prohibition by saying that it only applies with a beis hamikdash. Obviously, there isn't any harvesting of the O'mer nowadays, but the sha'agas aryeh holds that it would be assur until the morning of the 16th since the omer was able to be harvested all night after the 16th. The main proof of the sha'agas aryeh that the prohibition to harvest in E.Y. applies even when there is no beis hamikdash is that Tosafos in Pesachim 56a explains that the people of Yericho (mentioned in the mishna 71a) who harvested before the omer. The Sha'agas Aryeh understands from the juxtaposition to בית השלחין that the heter to harvest before the omer is that it is grain unfit for the o'mer, implying that otherwise it would be assur. Tosafos in Pesachim writes that the "anshei yericho" was after the destruction of the beis hamikdash, implying that even after the destruction there was an issur to harvest before the morning of the 16th from any grain that was technically fit for the korban omer. In the next siman (9) the sha'agas aryeh goes on to discuss whether if one were to harvest before the omer, the grain would be forbidden to eat.
The Rashash doens't mention the sha'agas aryeh, but quotes from the mishkanos ya'acov (y.d. 64) who also held that the prohibition to harvest before the 16th of Nissan applies nowadays. The Rashash aruges and says that since the pasuk makes it dependent on the harvesting of the omer, it will only apply when there is an omer to harvest. Tosafos 70b explains that the machlokes in the gemara between R. Yona and R. Yossi bar zavda whether it is assur until the omer is brought or until the omer is harvested, is speaking about the prohibition to harvest prior to the omer (not the issur to eat chaddash, as rashi explains). According to Tosafos approach, the conclusion of the gemara is that harvesting is assur until the harvesting of the omer, which implies that the prohibition is only when there is an omer to harvest for.