You don’t need more than a high-school science education to understand why cold fusion (at least the electrochemical variety) is very unlikely.
Calculate the electrostatic repulsion between two protons located about the width of a helium nucleus apart, using Coulomb’s Law. To find the equilibrium temperature needed for the protons to have that much kinetic energy (so when they smash into each other they can overcome the electrostatic repulsion and get close enough for the strong nuclear force to make them fuse), set the energy you get equal to kT, where k is Boltzmann’s constant and T is the temperature in kelvins. You can convert to degrees Celsius if you want, but the answer won’t change much.
One of the reasons the business persists among people who aren’t actual quacks is that there is a bit of a workaround: if you can somehow screen the electrostatic repulsion so it’s much shorter range – not less strong, mind you, but just dies off fast – then you can get the protons at a low temperature close enough that one can tunnel through the barrier. (This is sort of the reverse of radioactive decay, in which an alpha particle tunnels out of the nucleus at temperature far lower than would be required to overcome its attraction, via the strong force, to the rest of the nucleons. This works because the strong force intrinsically has a very short range.)
So how could you shorten up the range of the electrostatic force? A well-known process is electrostatic screening, which happens when opposite charges get between similar charges. The obvious example is a hydrogen (H2) molecule. In this case, two protons are located quite close to one another, such that their mutual electrostatic repulsion would make them fly apart at high speed. But between them are two electrons, and these two electrons ‘screen’ the repulsion between the protons. The physical process is that the two protons are mutually attracted to the electrons.
Can you get negative particles between protons, to screen the repulsion sufficiently to let tunneling take over? Not if you use electrons – we already know how close electrons can get protons, and the answer is the H2 bond length, which is not close enough to allow significant tunneling. (It does happen, however; every now and then an H2 molecule will collapse into an He atom when one proton tunnels into the other.) However, if you use much heavier negative particles – the classic case is negative muons – then you can get the protons much closer. So in fact there is such a thing as muon-catalyzed cold fusion: a stream of muons will form hydrogen-like “molecules” consisting of two protons and two muons, and the two protons will be close enough that one tunneling into the other is fairly common. So you have fusion, and at room temperature. The problem with this is that muons radioactively decay quickly, and can generally only be made by high-energy nuclear reactions, e.g. a nuclear reactor.
There are other approaches: you don’t have to rely on merely heating the protons to get them to the required velocity: you could, for example, accelerate them with an electric field. A potential difference of a megavolt should do the trick nicely. But how do you collect and direct through your apparatus enough of them to make macroscopic amounts of energy? You can also rely on the tiny fraction of protons that, in any room-temperature equilibrium, are already moving at fast enough speeds. Generally, it’s no real trick to make a few dozen H atoms per second fuse – and it gets you no noticeable amount of energy. The trick is to get 10^20 H atoms (milligrams of hydrogen) to fuse per second – now we’re talking power you can sell to the grid. So far, nobody has thought of a way other than simple heating plus compression (the compression makes collisions between the H atoms happen much more often, compensating for a low probability of fusion on each collision).
It doesn’t seem impossible that someone could figure out some clever nanotech way to manipulate protons to make them fuse. But what does seem unlikely is that anyone is going to do it with a macroscopic approach, like putting the hydrogen into an electrochemical cell. Pretty much by definition, any macroscopic approach has the system in thermodynamic equilibrium, and in thermodynamic equilibrium the rate of fusion is necessarily miniscule. Cold fusion can only be accomplished by some highly nonequilibrium process.