For instance, let's posit a character, sorc 1 wiz 1, that can cast Ice Knife through Sorcerer, and Burning Hands through wizard. But their INT is 16, and their CHA is 18. What would their save DC base off of, for Burning Hands?
Yes, it does
Your spell DC is specifically based on your class.
Sorcerers have the class feature:
In addition, you use your Charisma modifier when setting the saving throw DC for a sorcerer spell you cast and when making an attack roll with one. (PHB, p. 101, bold added).
While Wizards have the feature:
you use your Intelligence modifier when setting the saving throw DC for a wizard spell you cast and when making an attack roll with one. (PHB, p. 114, bold added)
These features apply both to single class sorcerers and wizards, and to multi-classed ones. As stated in PHB, p. 164 in the rules on Multiclassing (credit to Rubiksmoose for pointing this out):
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.
In this case, it would depend on one thing: whether you doubled up on spells. You would know two first level spells as a level 1 Sorcerer, but you only mentioned one. If you also selected "burning hands" as one of your two Sorcerer spells, then you could use your Charisma ability, and the DC would be 8+2+4=14. But if you didn't know "burning hands" as a Sorcerer, but only as a Wizard, then your DC would be 8+2+3=13.