I think that a moderate lifting routine can be a huge contributor toward general function and fitness on a tennis court. For me, the running around gets me better conditioned for, that's right, running around, but a whole lot of racquet swinging and trunk rotation can rack up some wear and tear.
Example: I tore and abdominal muscle while serving a couple winters ago and I'm pretty sure that it was because I'd been neglecting my work off-court for a couple of months. I've also been experiencing extra soreness lately in my arm, shoulder, and core, probably because I've been off my lifting routine.
I don't think that conditioning for tennis requires intense lifting and building a lot of extra muscle, but a reasonable workout schedule should certainly enhance both strength and endurance by a good margin. For me, that means I can swing at a ball over and over again with much less chance of wearing out and breaking down. I can play up closer to my potential.
It's usually easy for me to live with a lifting schedule where I take two days rest between lifting days. I also prefer to use only moderate weight with higher rep's so that the lifting sessions don't tear me up so much. I still get a good combo of strength and endurance from that sort of approach.
One note on general recovery: I've gotten into the habit of taking a few minutes to get a good basic stretch done immediately after I get off the courts - even before I get in the car to head home. Maybe I'll get one more stretch later on, but I've seen substantially better recovery and a lot less soreness and fatigue when I get that good stretch in right after I play.