The biggest effect I've see on resolution is due to plastic stress due to thermal gradients.

The higher resolution prints build up more layers of material, and each layer has a cumulative effect on thermal stress. The upper layers pulling up more as they cool, and the lower layers curling up more strongly as the layer count is increased.

To counteract this, a heated (or even just a draft free) enclosure makes a big difference. Having a heated print bed helps significantly, as long as the bed itself resists deformation (a sheet metal or PCB bed will bend more than glass under the same tension, for instance).

The actual plastic strength, however, appears increased. Laying down thinner layers of material appears to increase the bond strength between layers.

Resolution Drawbacks

With an ABS or PLA extrusion 3D printer, are there any potentially negative quality differences that could occur if I try to print at a higher resolution?

I am not concerned about print time as the equipment is not under high demand. I am, however, worried the device may be more prone to fracture, likely to have defects, or have other issues I cannot currently imagine.