I have an amaryllis plant that has nearly run its course and I want to replant it again. I'm growing it inside, so I have a year-round growing season. Once I cut the leaves and roots off for the bulb to grow again, do I need to let it be dormant for a period of time, or can I just replant it?
Yes, and no (see below), but it does it naturally, and there is no need to 'cut off roots and leaves'. You just leave it in its pot, allowing the foliage to die down, pull off the shrivelled, brown foliage, then just water very occasionally and sparingly so the compost doesn't completely dry out and shrink to a solid lump. When you see new growth starting, increase watering and move to a lighter position, if you've previously moved it while dormant to somewhere out of the way. It is, though, valuable to fertilize the plant while its leaves are still present, before it dies down, to help it produce good flowers the next time it grows. Use either a balanced fertilizer or one with a higher potash content.
It is possible to keep the plant 'evergreen', in other words, constantly growing, but this is more often done in hotter areas where the plant can reside outdoors.
Regarding the time span of dormancy, should be 8-12 weeks, when in theory, you could try to force it to start growing again, but I usually just wait for a little bit of green to start showing at the top of the bulb.
Some info in the link below, but it is a little confusing and not well laid out in terms of information, a lot of it pertains to growing outdoors, so needs careful reading!