Must a function that is decreasing over a given interval always be negative over the same interval?

No, the function doesn't have to be negative

Explanation:

We could have a function that is entirely positive on some interval where it is decreasing. The only requirement for a decreasing interval is that the function goes downhill as you move from left to right.

For example, the graph of y = x^2 is decreasing on the interval -10 < x < -3 but each y = f(x) value is above the x axis for this interval.