Is there some hierarchy of the human senses? What I mean by this is to ask if each of the sensory inputs can be quantified by the amount of information the brain is receiving: For example,

Brain says: currently receiving 10 packets of visual information, 3 packets of auditory info, 20 packets of somatosensory input etc...

Or maybe to put it in different terms, to measure the rate at which these sensory inputs are being recorded. For example:

Brain says: currently capturing visual info at 120 frames per second, hearing 20k cycles per second, somatosensory input at ...!?

And thereafter calculate the information content coming from each (per unit of time?) and thereby list them in some order.

Ultimately what I'm trying to understand/see is: if you had such a list (say, 1. visual, 2. touch, 3. hearing,....etc) and you began to remove modes of input from largest to smallest, would/might we begin to see weaker sensory inputs begin to enter the picture (sensory inputs that were being drowned out by the other senses)?
Now I think I have understood your mental model of human perception, I can give you an answer. Correct me if this is not what you meant to ask. 

If I understood you, you think that the human brain functions like a robotic brain. A sensor captures an image, sends it to the brain (which is comparable to a central processing unit), then the next one, etc. The brain then processes each incoming image. And analogically for senses other than vision. And you are asking if, if we stopped sending signals form one sense, the brain will have free capacity to process more informaiton from the other senses. 

The theory behind it

The problem is that you can't define it as neatly as that. On a very high level, this model is almost right, except for the part where the brain takes in one "captured frame" after the other and processes a frame as a whole. Technically, the eye really captures one frame (a fixation), does not see while it moves to a different viewpoint (a saccade), then it captures again. But from then on, the analogy breaks. 

The captured light is not sent to a central perception-processing place in the brain as a package. Rather, there are many neurons on the first step of the visual processing (V1 area). And each of them is connected to a few retina cells, a few fellow neurons from the same area, and neurons from the next step of visual processing. No V1 neuron "sees" the whole frame. Each V1 neuron will fire when it is excited enough; but exciting and inhibiting signals come from all directions - retina cells, neighbour V1 neurons, hierarchically "higher" V2 neurons - which means that the input for the object we perceived is not just the light falling on the retina, but also what we have just perceived. 

If you pick a single V1 neuron and count how often it fires, you can certainly count some kind of average rate, but it will not be much dependent on the rate at which eye fixations happen (which is actually a constant). It will have something to do with the amount of light  falling on the retina, and the rate at which this light changes, but these are only a few of the contributing factors - what the brain has already recognized is another factor, through the signals coming from the "higher" neurons. So, if you try measuring at the first stage of the brain - which is the V1 neurons for vision - you can't operationalise your measures. 

Besides, at that stage, no consolidation has happened yet. The information output of the V1 neuron is not a recognized object, they do very low-level recognition of textures, contours and similar. Information coming from other senses - the classical five ones, but also ones not widely known in popular culture, and ones which are known but not categorized as senses (such as pain, body spatial orientation) - gets incorporated into the recognized object along the way. 

But one thing to recognize here is that, if you are asking about the effect where the recognition process starts, then no, your idea won't work. The V1 neurons can't process information coming from the other senses. They are connected to the retina, not to the cochlea. If we amputate a person's eyes (good thing this is a thought experiment only), the V1 neurons won't start processing sound signals instead. 

But you are probably aware that blind people can recognize more of their environment based on other senses like sound, than sighted people wearing eyecovers. This indicates that the brain has learned to use the information coming through the other channels more efficiently. 

A speculation 

I think it is reasonable to assume that in the "sense switched off" situation the brain will also recognize more of the information coming in from other channels. The brain does not recognize everything that comes in from the senses, but only what it is interested in at the current moment. If there is nothing coming in through vision, it might "decide" to recognize more details coming in through sound than it would have otherwise. 

This means that the assumption you mentioned in the comments is true, if we start looking at the effect after recognition has happened - a person whose vision was "switched off" will extract more information out of the same amount of audio signals reaching the ear. But it is not really measurable, because we can only speculate how this information is represented within the brain. We don't have any metrics for "information as represented in the brain". Shannon found out how to measure informaiton which is stored somewhere accessible, as in a written word, but the information within the brain is not accessible, and its amount is only weakly correlated with the information available to the sensory organs. 

Be aware that not only is the amount of the information in the brain not the one available to the sensory organs, but it is also created from a mix of sensory perceptions and knowledge. At that point, you can't hold apart what came from the senses and what didn't. 


  When we look at something, perhaps 95% of what we consciously perceive is not what is "out there" but what is already in our heads in long-term memory. [Colin Ware]


Even if we could make measurements there, we wouldn't be just measuring the impact of the senses. We would have big trouble distinguishing it from the impact of memory.

So, can we have a hierarchy? 

I explained why we can't have a hierarchy at the input-end of the recognition pathways. I speculated that we can have a hierarchy at the output-end - but even if my speculation is true, we can't measure anything at that end. So, the answer to the question as a whole is: if there is a hierarchy, we can't measure it in the way you propose. 

I think there are experiments which try to make some rude sort of hierarchy, using observation of the kind "a person with covered eyes feels more disoriented than a person with stuffed ears". But this doesn't have the precision you are implying, does not answer to what extent the lower sense can be developed if the first one is missing permanently, and it can only be applied to the more prominent external senses like vision and hearing. (There are authors who argue that being certain of the truth of a statement is a sense - how do you turn that off?) Besides, experimentation with turning senses off can be dangerous, see research on sensory deprivation chambers for details. 



I don't have formatted citations for that post here. The information on the functioning of the visual pathway can be read in the book "Visual thinking for design" by Colin Ware. The info on how well one sense can replace another is certainly available in peer-previewed literature, but the one case I remember vividly is described by a journalist, http://www.mensjournal.com/article/print-view/the-blind-man-who-taught-himself-to-see-20120504. The author who claims certainty is a sense is Robert Burton.