This is silliness. Wearing a belt doesn't make your abs go to sleep and the rest of your body doesn't know you are wearing a belt. Your lower body is still lifting all that extra weight.
Whether or not you should use a belt depends on your training goals. If you are a competitive lifter, the goal is obviously to move as much weight as possible, and a belt unquestionably helps this goal. However, if you are trying to build strength for real world uses - such as sports, fighting, moving furniture - then a belt can be counterproductive for a few reasons:
1. In most weight lifting exercises such as the squat & deadlift, the torso must be kept rigid to transfer force to the bar. This is achieved through two methods: tightening of the core muscles (abs, obliques, erectors, etc) and intra-abdominal pressure from the Valsalva Maneuver (VM). A belt tends to only help the latter: by giving you something to push against, you can achieve much higher pressures and a more rigid torso. If you lift exclusively in a belt, it is possible that you will start to rely almost entirely on the intra-abdominal pressure and consequently not be working your core muscles nearly as much. In the real world, you are only as strong as your weakest link, so if your core muscles aren't developed enough due to using the belt too much, your strength won't transfer well outside of the gym.
2. Many people who use the belt too much get used to the proprioception and feeling it provides and aren't able to maintain safe/efficient technique without it. This is merely a mental block, but worth considering. I've seen people who keep their back nice and straight while deadlifting with a belt, but end up rounding their backs horribly while moving a couch, and injuring themselves.
The best bet for most people is to workout without a belt during the beginner & intermediate stages to develop a solid base of core strength and then do a mix of belted & unbelted lifting after that.