The NT doesn't record him being a Nazarite and as John the Baptist was a descendant of Aaron, it would not be permitted for him. See Luke 1.5:
It happened that in the days of Herod, king of Judea, there was a certain priest, Zechariah by name, of the division of Abijah. And he had a wife from the daughters of Aaron, and her name was Elizabeth.
On the other hand, one could argue that the prohibition only applied to those in active service as Priests and so would not apply to John the Baptist, but this is not the traditional interpretation of the prohibition on the sons of Aaron taking Nazarite vows.
UPDATE Why can't the Kohani leave the Holy place to hang out in the court of women with the Nazarites?
The ultimate answer is that for someone set aside at birth to be dedicated to the service of YHWH in his holy temple, to take on another vow, and a vow of a lay sect, would be viewed as a desecration and of violating the separate position the sons of Aaron have in the law and in the service of the temple.
That the Nazarites were a lay sect can be seen in Numbers 6.2 where it says "Speak to the sons/children of Israel" (women could take the vow) but it does not say
"“Speak to Aaron and his sons," as in Num 6.23. The sons of Aaron were made separate from the Israel -- that is their defining characteristic -- and this split of speak to "sons of Israel" (sons, being interpreted as group membership of the house of Israel not biological sons) followed by the "speak to sons of Aaron" was interpreted as rules to two disjoint groups, similar to "speak to the laity" and "speak to the priests". When both groups were the intended audience, it said "Speak to Aaron and to his sons and to all [the Israelites]".
Moreover thematically, the Nazarites were effectively a second priesthood, albeit a voluntary and limited time one, and were under a subset of the requirements that Priests were under, except that subset was stricter (Priests couldn't drink in the temple forever, whereas Nazarites couldn't drink anywhere, but in a limited time). Moreover the Nazarites had their own area of the temple (inside the women's court) whereas the Kohanim (sons of Aaron) served in the holy place, and the high priest entered the Holy of Holies on one day each year.
The nazarites could be viewed as a rival priesthood to the aronic one (most commentators believe the nazarite traditions preceeded those of the levitical priesthood -- see WBC commentary on Numbers), so you should read this passage in Numbers as also including a way to control and regulate the Nazarites by making them subject to and dependent on priests for all of the key ceremonies, thus cementing their subordinate role.
Note that not all Levites were of the sons of Aaron. It remains ambiguous whether Levites who weren't sons of Aaron could join, perhaps some of the talmudic scholars on here could clarify the traditions for non-priests, but the nazarite sect was viewed as something available to the laity, not to the sons of aaron, as these were left out in the address of Num 6.2