As David suggested, the normal function for a human brain when presented with a great deal of information is to drown out everything but a single focular point - you're probably already doing this while reading this question, ignoring sounds, ambient light and other informationt that is irrelevant to your current task.
But this is not just true about ambient information - you've surely heard of information overload and the very real (and current) problems it presents. This is a separate and entirely different phenomena from the sensory overload that David is describing - wherein the brain attempts to block out most of it. Though the effect is much the same.
Say you're reading an internet article, that one I just linked above for example. Your brain will try to process the information as you read it, but if you have say, multiple different articles open on your browser at the same time, and are attempting to take in all of the information they have at once, the information you get will be incomplete and fractured - the same as if you are having a sensory overload, your brain will try to reduce the input to a manageable level.
But say you don't stop, or literally cannot stop, and are forced to take in all of that information in a continual stream with no break or rest. Over time, parts of the information would get through to you, fractured, disjointed and without context. They would be lodged in your mind, and you may not even be aware of their presence. It could very badly damage your ability to think for awhile, or possibly even for prolonged periods, as sensory input in excess can damage your ability to take in information from those senses. Assuming it is an option, your brain may shut down and try to make you sleep due to mental exhaustion, but if you can't sleep, you may start to experience symptoms of sleep deprivation, ranging from mild to extremely severe.
In short, if you are forced to take in more information (not just sensory information, but actual information) than your mind can process at once, you could suffer severe mental breakdowns over time.