My mom is currently in a rehab facility but will be going into a nursing home soon. What will happen with her home taxes?

I'm sorry to hear of your mother's situation. The first question is: Does this appear to be a permanent situation? If mom will be moving from her home and her confinement will be long-term there are many financial issues to deal with and I would encourage you to explore long-term care financing issues and public benefit planning as soon as possible.
With respect to your immediate question I will assume you are speaking of property taxes. If mom will continue to own the home then of course property taxes will continue to be payable. The only question may be whether or not she will be able to maintain her "homestead exemption" if she currently receives one. I would call the property appraisers office or the tax collectors office in your county.
If she will be applying for Medicaid benefits there will be much to consider regarding the residential property as well as other assets.