In a math lab I worked at, faculty did one or two hours of reassigned time in the lab. This was very helpful during busy times if the faculty actually tutored. It also gave the tutors an opportunity to interact with faculty and develop a rapport with them. Students liked being able to access their instructions in the lab as well.
Some instructors would do study groups for their classes in the lab. As the faculty support, increased the morale of the tutors increased, due the feeling of working toward a common goal.
At a learning support center I worked at, they had faculty representatives for each department. This helped to centralize information and communication and gave the tutors a greater sense of being part of the educational process.
I hope this helps.