It really depends upon the intensity you use. If you are doing a low rep heavy weight routine and taking it to failure or serious fatique you will need the rest between sessions to let the muscles rebuild. If you are doing high rep, low weight and not going to failure you probably don't (if you are doing this routine with a lot of intensity you probably need the rest).
I always let my body tell me when it needs a rest. If I lift and the next day I don't feel that the muscles are tired I don't have a problem doing it again. If they feel fatigued, I let them rest. Right now because of time and lack of desire to be inside the gym, I lift 2 times a week for about an hour with excercises that hit all bodyparts - I can feel the muscle fatigue the next day but it is usually gone by the second day so I lift again.
The rule of thumb according to my trainers in college was to train heavy on two bodyparts one day, heavy on two more parts next day, and heavy on the last two on day three. Then do the same part split at a lighter weight, higher rep the following 3 days and rest completely one day. This is a bodybuilder type workout more than a powerlifter type workout.
Rest certainly doesn't hurt you if you are training hard. I know back when I was playing vball, I trained very hard to the point of overtraining. 3 hour practices were followed by a good hour in the gym - I loved to workout. When I quit the team (coach and I didn't agree on things), I didn't workout or play for about 2 weeks. I went to play on a vball team of other ex college players and my first jump to spike scared the heck out of me. I was up in the air looking down at the other team and realized I had my waist almost at the top of the net and I had to land!!!
Unfortunately, the super burst of strength didn't last many days because that's how muscles work. I had to get back to the gym. But the two weeks off certainly helped rest my overtrained body.
If you are overtraining, you will feel tired, lethargic, weak, and sometimes a bit depressed. Watch for the signs and take a rest if you see them. Most people can't overtrain.