How, if at all, would our perception of time change if the earth was shaped like a crumpled-up ball of paper?

How large of a difference in gravitational potential would there be between the peaks and troughs of the crumples, and how does that compare to the topology of Earth?

The same as normal Earth, same as you would compare a mountain to a valley. Essentially the same conditions we live under, but with a different landscape. What most intrigues me is the idea of a piece sticking out, a kind of land peninsula in space. Would one side be hours ahead of the other? I understand that this is highly unlikely, but let's pretend it happened.

 

Crumple up a piece of paper and you can see why I ask, the nooks and crannies provide for an interesting (although hypothetical) environment.

I see. The answer to your question is, yes... There would be a relative difference between the peak and the valley, but it's no where near as large as you imply. It's more on the order of a nanosecond than anything even closely resembling long periods like hours. In much the same way, we experience slight time dilation when flying on an airplane (see Hefele-Keating experiment), but it's so small as to be disregardable.