What the heck is quantum tunneling?
This is key to a coherent discussion of the question. Unlike the name suggests, the phenomenon does not necessitate a repeated interaction, nor does it involve the weakening of the potential barrier. Quantum tunneling is a phenomenon well accounted for in modern quantum theory, in which the phenomenon is built-in mathematically.
But it is the wave like nature that allows matter particle to overcome physical barriers, physical barriers here could be anything like electric field you name it.
The issue with this view is that it incorrectly assumes how modern physics views the quantum world. Particles, as described in quantum mechanics, do not have a "wave-like" nature in the physical sense. The Wavefunction describes the probability of each of the possible "states" the particle may be in.
The wave aspect here refers to the built-in limitations on our knowledge of how the particle is going to interact, which is described in probabilistic terms that have the same mathematical form as are those used to describe physical waves (note this does not mean that they are waves, analogous mathematical constructs are common in physics and are never understood to imply a physical connection).
Physics does not describe the electrical potential as forming a "physical" barrier. Instead, the "electrical field" is a mathematical description of the electrical interaction. Now the above distinction may appear pedantic, however, as we shall see it is critical to a proper answering of the question.
Now suppose something weird is happening right now, every ongoing experiments showed no sign of tunneling regardless the size of barriers... OMG quantum tunneling is broken! What the worst that can go wrong beside having to reprint all the physic textbook?
Since the mathematical theory of quantum mechanics necessitates quantum tunneling, if this phenomenon were to suddenly "turn off", either the theory would change, or math would break.
If the theory were to break, the probabilities would change and this would have consequences well beyond the phenomenon of quantum tunneling, instead the entire theory would fall apart. From our theories therefore, it is impossible to say exactly how things would be affected since we would have to reformulate the mathematical theory because the old one becomes broken.
In other words...unless you have a very good understanding of physics (and even then), it is almost never a good idea to modify a physical theory (or even just an equation) and work forwards to the consequences...they will certainly be too numerous and consequential to coherently discuss.
Of course one way to approach the question is to look up all the phenomena that include the phenomenon of quantum tunneling and assume these phenomena wouldn't happen anymore. That is, however, naive from a physical perspective, to understand why it is necessary to look a little at the mathematics of quantum mechanics vs. classical mechanics.
Classical mechanics views all interactions through forces (which can be characterized mathematically by energy potentials, this will become important in a minute) upon "particles", which at the simplest level would be would be described by non-deformable masses which fill a definite volume of space. The most pertinent restriction on how forces and particles interact in classical mechanics is described by the conservation of energy: $E = T + V$. For a single particle the total energy is the kinetic energy possessed by the particle added to the potential energy imposed by the forces acting on each particle through each force's energy potential.
Quantum mechanics, on the other hand, does not view interactions like this. A "particle" is a mathematical description which describes localized "states" (or collections of information), the best we can do is describe this localized collection of states through guessing information. Our guesses are not random however, we educate our guesses through probability and statistics, which has been found to follow the following equation: $E \Psi = (T+V) \Psi$. The difference is apparent, for a single particle we describe the probability of, say, energy. We first specify a given state. For a given state we can compute a particular energy. The states a particle may have are determined by through the wavefunction. Of course this doesn't mean that a particle can have any energy value, or be anywhere. Take for example the (relatively) simple case of the hydrogen atom which is a classic (pun intended) problem in an undergraduate course on quantum mechanics. The electron's wavefunction (and corresponding probability densities shown as spatial distribution across space) is shown here.
To understand this idea of multiple states, think of a ball in a normal sized bedroom. Suppose that you had to try to figure out where the ball was. Now the ball could be anywhere, right? Well, if suppose you knew the owner of this room and they were quite tidy, you could guess that the ball is more likely going to be in the closet and not lying in the middle of the room.
Quantum tunneling is no different. We cannot know definite things about a particle, for instance, its momentum. When an interaction occurs, we can compute the likely energy of the particle. In our analogy, we might find that the ball is in the closet 99/100 times randomly checking throughout the day, or a particle in a nucleus has an energy of -8 MeV, 99.999% in a given second. However, this is not definite. The ball might end up in the middle of the room during the time the owner takes it out of the closet to juggle for fun, or the particle might have 0.8 MeV of energy and fly out of the nucleus.
Thus, since quantum tunneling is an inherent part of quantum mechanics, if this phenomenon were to suddenly disappear, quantum mechanics would falls apart and anything goes.