Another way to write your sentence is:
If you need anything, then let me know.
Let is being used imperatively. Intuitively one would likely associate this to present tense, making this a "zero conditional." But will also can be used to suggest an imperative.
If you need anything, then you will let me know.
But this isn't really future tense, even though it looks like it
Using this site as a reference, the difference between zero and type 1 conditionals is:
In type 1 conditional sentences, you can also use modals in the main clause instead of the future tense to express the degree of certainty, permission, or a recommendation about the outcome.
And that's what you're doing when you add will above.
Other modals work, to express the degree of "certainty" ... of imperativeness:
If you need anything, then you { should | would | could | must } let me know.
If you're going to use this "zero, first, second, third" model of conditionals, it seems imperatives straddle both zero and type 1.