Assuming a patient and a client wanted to begin a romantic relationship, how long must they wait once the professional relationship is over? 
It would be unethical ever. The therapist must leave the door open to the possibility of the patient’s return to therapy.
Consider also that the patient and therapist have a relationship for a while and then the therapist wants to end it. The patient internally will almost always see this as rejection not only by a lover, but also by the therapist. That can be very, very damaging.
It is common for the patient, especially, to be a little bit in love with the therapist. And probably sometimes the reverse is true. But I just can’t imagine a good therapist acting upon those feelings or allowing the patient to. I think that to do so can probably get the therapist professionally busted.
Each state may have their own rules for their Licensed Professional Counselors. In Texas, it is not allowed:
Although ethically wrong, there may be a loophole so that the LPC can avoid getting disciplined:
Here is the link if you want to investigate further- remember, this just applies to Texas: http://www.tdh.state.tx.us/hcqs/plc/lpc/lpc_ethics.pdf
Question: A legitimate, licensed therapist would almost certainly know the answer to the original question. Is the poster a client interested in his/her therapist? What is the therapist saying about the idea?
Or the poster could be writing a novel. Or the poster could be curious about something they’ve read. Or the poster could know someone who is in love with their therapist. There’s no reason to think this is anything more than a hypothetical situation.