Ramban

THEN THOU SHALT TAKE OF THE FIRST OF ALL THE FRUIT OF THE GROUND, WHICH THOU SHALT BRING IN FROM THY LAND. [The meaning thereof is] that you shall take of the first of all the fruit which you will bring into the house from your Land, which the Eternal your G-d gives you. He commands that one should set aside that fruit in the field and designate it as first-fruit and then bring it into his house and put it in a vessel fit to be taken to the Sanctuary [in Jerusalem]. Now, Scripture did not prescribe an amount for the first-fruits, instead, even one grain [or berry] of that species exempts the entire field, similar to the law of terumah (the heave-offering) where one grain exempts the whole pile of grain.
AND THOU SHALT GO UNTO THE PLACE WHICH THE ETERNAL THY G-D SHALL CHOOSE. In line with the plain meaning of Scripture he admonishes against bringing first-fruit in [any] one of the cities [except for Jerusalem] after the Sanctuary was chosen [i.e., after Jerusalem was designated by the prophets and the ark of G-d was brought there], for just as He warned against slaughtering offerings outside [of the Sanctuary Court], so he warned with reference to the first-fruits. But the obligation of bringing first-fruits became binding immediately after they conquered and divided the Land, (above 12:9), because the structure of the House of G-d was not of stone as it had been at Shiloh; instead, it was made of the boards and curtains from the Tabernacle in the wilderness. Therefore during the fifty-seven years when the ark of G-d was first in Nob and later in Gibeon, it was permissible to bring offerings to G-d on the bamoth, “high places,” or private altars. See also further, Note 6. in accordance with the words of Rashi. And in the Sifre it is stated: 298. “And thou shalt go unto the place which the Eternal thy G-d shall choose to cause His Name to dwell there — this refers to Shiloh and ‘the Eternal House’ were permitted after its destruction. [in Jerusalem].” Perhaps the Rabbis [in the Sifre] meant to say that the bamoth (“high places”) were forbidden during both periods [i.e., Shiloh and Jerusalem]. However, during [the periods of] Nob and Gibeon [when high places were permitted] first-fruits could also be brought [only there in Nob and Gibeon], but they could not be offered at a private bamah, for Scripture refers [here in the chapter on first-fruits] to the altar [and he shall set it down before ‘the altar’ of the Eternal thy G-d indicating that only the public altar is suitable for first-fruits].. See further Vol. III, p. 123, Note 122 and Vol. IV, p. 180, Note 138. But since it is written, The choisest first-fruit of thy Land thou shalt bring into the House of the Eternal thy G-d, they perhaps did not bring first-fruits when it [the House of G-d] was in a tent and in a tabernacle but only in Shiloh where there was a stone building and in the Eternal House, were permitted after its destruction. [thus excluding the period of Nob and Gibeon].