"Do you have a fever?"
Sorry, I’m not quite sure which topics are already taught by this point… Let me try again, this time assuming a little less prior knowledge:
So 发烧 is a verb “to develop a fever” right (to be exact, it’s a verb group 发 (in this case) “to develop” + an object 烧 “heat, fever”)? So to say “I have a fever”, you would say: 我发烧了 (literally “I have developed a fever”). The 了 indicates that a change has occurred.
But here’s the thing about 了: If you want to negate a sentence with 了, you can’t use 不. Instead, you have to insert an auxiliary 没有 (or just 没 also works), and no 了. So “I don’t have a fever/I haven’t developed a fever” is 我没(有)发烧。
However the fact that you (in most cases) can’t use 不 with 了 not only affects the negatives themselves, it also affects the question construction where you put the verb next to its negation (which usually comes out as V不V). So you have to use the auxiliary here as well: Instead of V不V, you get 有没有V “have or haven’t V”.
Thus, the V-negation-V form of 你发烧了 is 你有没有发烧 “have you or haven’t you developed a fever”. 发烧 is still a verb here, not the nominal object of a full verb 有 “to have”.
Maybe it’s clearer with a few parallel examples: - 你有没有去邮局？ “Have you gone to the post office?” - 他有没有拍照片？ “Has he taken [any] photos?” - 你们有没有吃午饭？ “Have you eaten lunch?”
2020-06-24 你发不发烧 is accepted. But based on AbunPang's replies here, I think it is not as specific a question. 你有没有发烧 specifically asks if you have already developed a fever. 你发不发烧 could be asking if you got one in the past, are getting one now, or even if you will get one later.