Introduction
Between the main rulebooks and the Magic Burner supplement, Burning Wheel contains rules for roughly a dozen different types of magic. Not all of these are intended to be used together, and not all are appropriate for the Hârn setting.
Below are my suggestions for which specific BW magic rules to use and how to use them.
Sorcery
Burning Wheel's Sorcery rules
are ideal for the spell-based magic of Hârn's Shek-Pvar mages.
If desired, the Abstraction and Distillation rules from the Magic Burner can be used for improvised magic and spell creation in conjunction with the basic Sorcery rules from Burning Wheel Gold.
Please see the Shek-Pvar page for more ideas on how to adapt Sorcery for Shek-Pvar wizards.
Faith
Burning Wheel's Faith rules are recommended for all miracles, invocations, requests for divine intervention and other forms of 'religious magic' in Hârn.
The Religion rules (from the Magic Burner) can be used to supplement the basic Faith rules. More ideas can be found in the Faith pages.
Elven Songs & Dwarven Arts
Burning Wheel rules provide Elves and Dwarves their own unique forms of magic: Elven Spell-Songs and Dwarven Arts.
Although canon publications suggest that Hârn's Elder Peoples are familiar with Pvaric magic, I recommend using BW's Elven Spell-Songs rules for Sindarin magic and Dwarven Arts rules for Khuzdul craft instead of the Sorcery rules. They add color and flavor that are appropriate to the characterization of each race in Hârn.
Other Possibilities
It would be possible to use the rules for Art Magic (pp. 70-80 of the Magic Burner), instead of Sorcery, for Pvaric mages.
If rules for the creation of items of power by mages, alchemists, etc. are desired, the Enchantment rules can be used.
Burning Wheel's rules for Spirit Binding could potentially be used to represent the religious magic of Hârn's barbarian tribes to speak with and command local/place spirits.
The rules for Death Art and/or Corruption could potentially be used to represent the creation of Amorvin and the influence of the Shadow of Bukrai within the Morgathian Church. Blood Magic could similarly be used in connection with Faith to represent represent the ritual torture and sacrifice conducted within the Agrikan church.
Pages exploring these possibilities may be added later.
Not Recommended
Some of magic types outlined in the Burning Wheel rules are not good matches for the Hârnic setting. In particular, Summoning and the rules for Familiars and Homunculi don't quite seem to fit the mood and flavor of Hârn.
The rules for Orcish Rituals of Night and Blood are probably also inappropriate given the purely non-magical nature of Hârn's Gargun.