The GameBoy can change the volume of both of it's square waves without resetting, however, only square 1 can do note and hardware sweeps. The digital channel has only 4 volume settings, 100%, 50%, 25%, and 0%, so there aren't really many crazy things you can do with the volume. You can have it sett to a particular level, or have basic sweeps from "100-50-25-0", oscillations like "100-0-50-0-25-0" or basic stuff like that
In normal operation if you change the volume, it resets the duty cycle every time it changes. This produces fun clicks (unlike the NES). There is a "zombie mode" you can use to make it act like the nes (no clicks/phase reset) but it is buggy or something from what I remember.