You'd be unlikely to go to jail for not paying your lawyer.
If your lawyer is still working with you on a case and you haven't paid the lawyer then they would most likely just withdraw from the case and stop being your lawyer.
If the lawyer is already done fighting for you in court and then you fail to pay the lawyer then the lawyer could take you to court and sue you for the money.
But it's unlikely that you'd be put in jail for not paying a lawyer.
Lawyers have malpractice insurance in place and if the lawyer sues the client for money then it could actually affect the lawyers insurance rates and make them higher so it's usually in most lawyers best interests to just withdraw from the case and not sue you.
However the lawyer still may sue you once the statue of limitations for malpractice runs out.