The only requirement is that vehicles on public roads be registered in a state. The rules and requirements for registration vary by state. Suffice it to say that if you can legally keep it registered in any state you do not have to change the registration based on where you live or operate it.
Residency is often a requirement of registration, so if your are not a legal resident of the state of registration you may not be allowed to keep it registered in that state.
You may also be required to change registration due to insurance. For example, in some circumstances an old insurer won't be able to sell you insurance if the car is relocated to another state for an extended period, in which case the requirement to have the vehicle insured could effectively force a change in registration.
Amendment: Nate Eldredge points out in the comments that some states do require residents to register their cars in state. I expect this is more common in states that derive significant revenue from vehicle registration, and that enforcement would be commensurate with the size of that tax.