"sy" is a special part of the verb "to be", used in focused sentences. Without it, the sentence has no verb, so yes, it needs to be there.
Remember, this isn't a literal translation of the equivalent English sentence, it actually means something like "Which colour hair is with you?", with the "sy" corresponding to "is".
I have about as much experience with sy as any of you, but I think you can sort of look at it like atá in an is construction in Irish, where it's sort of the connector that makes the sentence make sense. Without sy, this sentence is "what color hair with you?" but the sy acts like is it that is, so it's more like "what color hair is it that is with you?" It turns it into a real sentence, and gives the prepositional phrase gyda ti something to connect to.
"Sy(dd)" is the question form of 'to be' ("bod") when the question word or phrase is the subject (source: King 2003, p. 100-102). That's the case here: the literal translation is something like "what hair color is with him", meaning that the question phrase, "what hair color" is the subject.