Looking at the article you linked, I think I can give a meaningful answer.
IIT argues that feed-forward systems are not conscious. This means any system which can be well modeled as an open loop that accepts inputs from one side, and provides outputs on the other side, must not be conscious because all of the information it integrates is available just by processing its inputs. Systems with feed-back function differently. Systems with feedback can observe the effect of their own actions, and adapt accordingly.
When they talk about a digital computer being minimally conscious, what they are most likely referring to is a traditional brain simulation, where you have a body of canned inputs and observe its outputs. Even if the system has feedback loops within itself (which a brain simulation would), the entire system has limited consciousness because it can only integrate the inputs that it receives. If the canned inputs contain very little data, it is hard to generate a large amount of integrated information about them (information cannot be "created" out of thin air).
In IIT, the concept of the environment is very important. When you get down to the actual mathematics to describe the Phi function of a system (measuring its level of consciousness), it is dependent on the ability to exclude states which are theoretically possible by the model, but cannot actually be arrived at by the entity under test. The fundamental measures of information used in IIT are literally information about a system. The classic example is the Maxwell's Daemon. You have a system with two boxes connected by a door. One gas particle bounces around in the system. If you have information that the particle is in one box, you have reduced the number of possible states of the system by 1/2, corresponding to 1 bit of information about the system. In IIT, Phi ends up measuring how much information the conscious entity can generate (in bits) minus the amount of information that could have been gathered simply from the input (in bits). Clearly, for a feed forward computer, virtually all of the information about the system is available in the inputs, so it is not very conscious. In the case of a brain simulation with canned inputs, the state of the system is fully knowable with the information at the onset (its fully defined by the blob of canned information plus the code of the brain simulation).
Connect the brain simulation up to the enviornment, however and the rules shift. Now there are unknown variables in the system, because they are interacting with the physical world. Now, if these unknown variables are affected by the computer's actions, and observable by its sensors, we have the capacity for feedback. The computer can try something, and observe the effect of its own actions. In doing so, it can begin providing information that was not known at the onset of the simulation -- information that has grown over time as interactions occur.
Now if we were to snarf all of the data for this simulated brain as it crosses the analog to digital barrier, and saved it, we would generate a set of canned data again. This computer that was conscious is, by the rules of IIT, rendered virtually unconscious because we have a body of data which can be replayed to get all of the information that simulated brain had. However, to declare it as thus, we have to capture all of its inputs. In real life situations, data capture like this is difficult, quickly consuming terabytes of data! So, in practice, it is often not reasonable to capture every single bit of digital input to the simulated brain. Those bits now have to be thought of as unknowns in IIT, and since they are unknown, we can give the computer credit for "knowing" information that is not available in any other pile of data besides "itself."
Now, consider the internet. The internet is massively parallel, and it is this parallel-ness that is important for why it might be conscious. Even given a body of canned input, the internet could still be conscious because its internal state is dependent on a remarkable number of timings. If one part of the internet is fed data that tells it to read a block of data from a server, while another part of the internet is fed data that tells it to change that block of data at the same time, slight differences in how fast those parallel processes execute lead to changes in behavior. These changes in behavior were not knowable from the canned input -- they are emergent as a result of slight analog effects such as clock skew. Thus, the internet contains the integrated information of a billion small transactions which are not captured in any database anywhere, but the result of those transactions can be fed back into algorithms in parts of the internet. This allows the internet to demonstrate consciousness, by IIT.