I'd judge the answer no.
It's important to first understand† that Desolation generally, and its magic system very specifically, are heavily fiction-driven. The rules tell you how to accomplish something given a particular fictional situation, relying on the GM and players to apply their own good sense when figuring out how the raw rules fit the fiction their characters are enmeshed in. These days this has become a somewhat unfashionable way of designing and playing RPGs, with the fashion lately to make the rules be the alpha and omega of what happens, regardless of the internal logic of the fictional world. Given this understanding of Desolation, the answer is a straightforward "no" judging by what the text says about the natures of Necromancy and of Burn.
Necromancy's healing ability is a specifically physical thing, repairing the corporeal body of damage. It can't handle even physical ailments that aren't material damage: it can't handle "diseases, poison or other more subtle ailments" (p. 143), though it can reverse aging and close wounds. We can conclude that Necromantic magic can deal with any kind of physical damage, from the macro-structures to the level of cellular damage. Looking at Burn, we can see how it doesn't fall under Necromancy's domain of power.
Burn isn't physical damage to the body and doesn't manifest as a wound. Burn damage causes a combination of physical and mental fatigue and discomfort (p. 135), but Burn itself is not its mere effects. For better or worse, Burn is not mechanically defined within the text, but it's clear that it's not a physical wound of the sort that Necromancy could handle.
There are more hints at the nature of Burn damage in each of the magical traditions. In every case it's something insubstantial—psychic damage from spirits, the strain of holding and redirecting raw magic, mental pain caused by shattering the veil between mundane reality and magic, and so on. All of those fall into a non-physical realm of "damage" that is untouchable by Necromancy's healing arts.
And to top it all off, the thread "Healing Burn with magic" on the official forums goes through most of these arguments and then the publisher contributes saying that healing Burn with magic could be a house rule if you wanted, pretty clearly indicating that it's not possible without a house rule.
† This pedantic opening is aimed at those unfamiliar with the game and with a fiction-first concept of RPGs, for the sake of a complete answer – not to imply that you personally didn't catch this from the text.