My mom is in a nursing home as a Medicaid resident. I have her car at my house. How long do I have to keep it, or can it be sold?

Whose name is the car in? Only the person on title can sell the car. Otherwise it becomes a part of Mom's estate unless your Mom is able to go to DMV (Triple A offices do this as well if you are a member) and change title. Immediate family can be put on title. I would also see a lawyer before you do something catastrophic such as having a Mom on Medicaid, but selling an auto in her name, hence earning her money, and ruining her medicaid allottments OR having Mom sign over a car and having it be considered gifting and ruining her medicaid.
I would be very careful. We are a Forum. Legal advice is what you need now. If this car has any value it may be worth selling, but if it has any value you may be up against severe and serious repercussions in messing with a title you aren't on, that could ruin your Mom's medicaid until the amount the car is sold for is paid for Mom's care.
It is quite likely that it is best just to let this car sit until it becomes a part of Mom's estate.
Who is paying the insurance on this car? Another question that figures in all this.
See an attorney. The 350.00 for the hour of Elder Car Time is worth it and given I have heard recently of attorneys who won't see someone without a 700.00 be certain this attorney will work by the hour to answer your questions re the car, the title and your individual situation.
If Mom is capable of putting you on title with her, you can use this car and sell your own? Lots to think about but be certain you get legal advice for your own area of the country.

I sent you a PM with some info.

If u sell it, gives Mom more money than allowed if the proceeds go over the asset cap. Could effect her Medicaid.