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After our party slew a frost dragon, my character harvested a few dragon teeth off of the corpse. I now want to have someone craft weapons out of those teeth (DM does allow crafting of magical items.)

Would these dragon teeth be inherently magical when crafted, or would they need to be imbued with magical effects?

As a note, this is within a game of Storm King's Thunder, so the setting is Faerûn.

There is nothing currently in the 5e rules about harvesting dragon parts. You therefore need to work with your DM on any effort you make at crafting something unique and useful from those teeth.

Our party is using a 3.5 version of the Draconomicon that talks about harvesting parts. From what I recall, the teeth are not inherently magical. I get that from there being no mention of any part of the dragon being magical except for the draconis fundamentum which powers the dragon's breath weapon. Although in 5e, dragon's breath isn't magical.

Dragoncraft items are nonmagical objects made from specific parts of a true dragon’s body. Only a character with the Dragoncrafter feat (see page 105) can create dragoncraft items. They derive special powers from their origin, as well as from the skill of the person crafting them.

Dragonfang Weapon: Dragonfang weapons are masterwork weapons crafted from the claws and teeth of a dragon. In addition to the +1 nonmagical enhancement bonus on attack rolls granted by its masterwork quality, a dragonfang weapon deals 1 point of energy damage on each successful hit. The type of energy is the same as that of the dragon’s breath weapon... This damage is treated as an extraordinary (and thus nonmagical) feature of the weapon

(Draconomicon, Pages 116-117 - thanks Raj!)

As with any part of 5e, your DM is the final arbiter in what can and cannot be done.

There is no text in the 3 D&D 5e core rulebooks that would indicate Dragon parts contain inherent magic.

It would be worth reading the section of the DMG that discusses crafting magic items. Under the heading Crafting a Magic Item the DMG (p 128) states:

Magic items are the DM's purview, so you decide how they fall into the party's possession. As an option, you can allow player characters to craft magic items.

The creation of a magic item is a lengthy, expensive task.

This would suggest that you can't tie a dragon tooth on the end of a stick and call it a magic spear, or sharpen it and place it in a handle and call it a magic dagger. Magic item crafting is generally done during downtime when your PC has a lot of time to dedicate to the craft. Additionally in this same section the DMG states:

To start, a character must have a formula that describes the construction of the item. The character must also be a spellcaster with spell slots and must be able to cast any spells that the item can produce. Moreover, the character must meet a level minimum
determined by the item's rarity, as shown in the Crafting Magic Items table.

The dragon tooth could certainly be used as an ingredient and could potentially give some qualities to the magic item, however according to the DMG crafting a magic item takes a lot of time, and needs specific GP and class level depending on the rarity of the item.

Also note that in most cases the formula to create a magic item is more rare and more valuable than the item itself. A sidebar on magic item formulas in the DMG (p 141) says this:

You can award a formula in place of a magic item. Usually written in a book or on a scroll, a formula is one step rarer than the item it allows a character to create. For example, the formula for a common magic item is uncommon. No formulas exist for legendary items.

The text goes on to state that even in a world where magic is easily available the formula is at least as rare as the item and the GP value for the formula is 2x the item value.

As is often the case in D&D 5e the details of how magic items are crafted in your world are left to the GM's discretion, though the DMG does offer some guidance.