In the Tanakh, there are two words for razor for shaving. The more common one, תַּעַר ta'ar, is found in five verses: Bamidbar 6:5, 8:7; Yeshaya 7:20; Tehilim 52:4, and Yechezkel 5:1. (In Yirmiyahu 36:23, it refers to a scribe's knife.)
The less frequent word, מוֹרָה morah, only appears three times: Shoftim 13:5, 16:17; and Shmuel I 1:11. All of these mentions of morah refer to nazirites (Shimshon and Shmuel).
The laws of the nazirite are found in Bamidbar, and that is where ta'ar appears. The phrasing of the verses is very similar. Bamidbar 6:5 says תַּעַר לֹא־יַעֲבֹר עַל־רֹאשׁוֹ "no razor [ta'ar] shall touch his head." Of both Shimshon and Shmuel the verses say וּמוֹרָה לֹא יַעֲלֶה עַל רֹאשׁוֹ "and no razor [morah] shall come on his head." This would seem to indicate that the two words are synonymous - referring to the same object, first in the law of the nazir, and then in the stories of two nazirites.
This understanding is reflected in the etymology of the two words. Many recent scholars say that they share a common origin. For example, Klein writes in his entry for morah:
Of uncertain origin; possibly contraction of מַעֲרָה, from ערה ᴵ (= to lay bare), whence תַּעַר (= razor).
The root ערה, "to lay bare, strip" (the source of arom ערום - "naked") therefore led to both words. Morah was a contraction (the ayin dropped out) of ma'areh (meaning an open, bare place - see Shoftim 20:33), and ta'ar was a different way the noun was formed.
Kaddari also accepts this theory, and expands it by noting the connection between the root גלח - "to shave" and גלה - "to uncover, expose."
This same root - "to reveal" - can explain another usage of ta'ar in the Tanakh. It can also mean "sheath (of a sword), scabbard" (Shmuel I 17:51; Shmuel II 20:8; Yirmiyahu 47:6; Yechezkel 21:8,9,10,35). As Klein points out:
Prob. from ערה ᴵ (= to lay bare, uncover), whence also Ugar. t‘rt (= sheath of a sword); hence of the same etymology as תַּעַר ᴵ.
Gesenius says that ta'ar as sheath, "perhaps so called from emptiness."
Today, morah is almost never used for "razor" (probably because its other meaning, female teacher, is much more prevalent). Ta'ar is used for razor, although the phrase סַכִּין גִּלּוּחַ sakin giluach is also common. As far as sheath/scabbard, I guess I never had a reason to use the word, since the current word surprised me: נָדָן nadan. But it too is biblical, found in Divrei Hayamim I 21:27. However its origin is Persian (see a discussion here).