Motorized scopes don't have this problem...
Depending on the type of mount, they might have the second order problem of field rotation. For properly aligned equatorial mounts the telescope tube is also rotated about its axis when it is rotated around the polar axis of the mount, so for example north is always pointing in the same direction at the focal plane of your camera.
For telescopes on alt-az or "azimuth" mounts this doesn't happen, so on the scale of hours any object that's not in the center of the field of view of a properly aligned telescope will execute a circular arc around the center of the exposed field.
What is the optimal exposure time for photography with an unmotorized telescope?
It depends on what you define as "exposure time". There's total integrated time used to produce your image which can be quite long, and your frame rate, how frequently you dump one "exposure" to memory and start recording a new one, for later recombination into a single final image.
Most astrophotography both pro and amateur relies on the latter technique to avoid overexposing bright objects in the field that might only need a few seconds or minutes to saturate, while still capturing dim objects.
Saturation can cause some CCDs to produce other artifacts in other areas of the frame, so you really do want to avoid it.
So if you don't have tracking, you may need to make your frame rate of the order of one per second or even less, depending on the scale and resolution. I think other answers here will go into more detail on that, or at least link to other questions and answers here that already address this.
Note: Once you are making many shorter exposures, you can then pause to move your object back to keep them near the center of the field at regular intervals. The software you use to combine images will do the realignment as long as there is not appreciable field distortion. If there is, then you'll need to work harder to keep your object centered, or include a complicated model for it in your image-combining software.
Modern DLSRs, astrophotography cameras and even some cell phone models will have ways to program regular frame dumps for image accumulation.
There are numerous programs you can buy or get for free to reassemble those images into a single final image. These automatically align all of your exposures before adding them, and have features to lower the resulting noise.