In the movie Interstellar, they go to a planet where time flows by much slower for them while on it because of the increased gravity that planet is experiencing due to the nearby black hole. My question is whether or not something much smaller, like an asteroid or a dwarf planet even, could pass by Earth close enough to slow down time from our perspective? Is there something that could do the opposite and speed it up, and effectively slow everything else down?
Time wouldn't slow down from our perspective. The time dilation would be evident only if we were to check our clocks against a clock that's far enough from the celestial object (just as shown in interstellar) and/or after the celestial object is gone.
That being said, since it's a function of mass, a sufficiently massive asteroid could cause dilation significant enough to be observable from one side of the planet to the other. But if the asteroid is too massive (because a millionth of a second off here and there won't make an interesting plot point), there might be many other inconvenient (and potentially devastating) effects of such a massive object passing by. How a small asteroid/planet is able to be contain so much mass would probably require a super dense fictional substance. Alternatively, you could just speed up earth, but it'll mean that the time dilation is same everywhere on the planet (aside from other interesting/devastating consequences).
Speeding time up on earth in comparison to rest of the universe is much trickier but if we were to speed up the time on earth compared to another planet or a region of space then it's doable - Just put the massive celestial object close to the region of space and you could, in theory, say that the time on earth has sped up compared to that region of space. The region must be far enough from earth so that earth is only nominally affected by the massive object.