I'm not a lawyer, so this isn't legal advice. For that, you need a lawyer. However . . .
No, it's not illegal for a landlord to have a tenant sign a quit claim deed. A quit claim deed says that the person signing it (the tenant in this case) gives up any ownership claims he/she might have on the property. If the tenant doesn't own the property or any portion of it, the quit claim deed is worthless.
You could sign a quit claim deed for the Statue of Liberty. It could be duly witnessed and notarized. But since you don't own the Statue of Liberty, the quit claim deed is worthless. You're saying you're relinquishing ownership in something you don't own. So?
As far as property taxes, the owner of the property is responsible for the property taxes. Thus, you have Owner A and Tenant B. Tenant B signs a quit claim deed giving up any rights he has to the property; he has no rights so he's not giving up anything. Owner A is still the owner--always has been--and is responsible for the property taxes.
Now . . . if Owner A signed a quit claim deed giving his property to Tenant B AND if Tenant B filed the document with his local authorities, then Tenant B could be the new owner of the property. In that case, the new owner (former Tenant B) would be responsible for the property taxes. But that's not the scenario you described.
For more information, see a lawyer.
Hope that helps.