I have read that the fine structure constant may well not be a constant. Now, if this were to be true, what would be the effect of a higher or lower value? (and why?)
Another thing that would be changed by a varying fine structure constant would be that it would alter almost every electromagnetically mediated phenomenon. All of the spectra of atoms would change. What would also change would be the temperature at which atoms can no longer hold onto their electrons, since the strength of attraction between electrons and the nucleus would change. This would then change the redshift at which the universe becomes transparent. The end result would be that the cosmic background radiation would be coming from a different time in the universe's history than otherwise thought. This would have consequences for the values of cosmological parameters.
Once you alter phenomena in this stage of the universe's history, though, you have to be quite careful to not disrupt the current predictions for how much hydrogen, helium, and heavy elements there are in the universe (while creating nuclei depends mainly on the strong interaction, electromagnetism does have something to do with determining the final energies of the nuclei, and so can't be completely neglected--changing the fine structure constant changes these cross-sections). Current theory predicts these things with great accuracy, and changing things around, particularly particle physics parameters that govern the length of the neucleosyntheis era (which overlaps with, but is a subset of the time at which the universe is opaque) potentially make these observations not agree with theory.
We see the same spectral patterns arising from stars, galaxies, and dust clouds very far into space (which is to say very far into the past), and these features are dependent on the fine structure constant.
So, yes. At least to a first approximation.
There have been attempts (by reputable scientist) to form cosmologies to compete with the current big-bang-and-inflation scenario, and wondering about the constancy of various fundamental "constants" was one of the tools that people used in trying to form them. I don't know the exact state of any of these efforts right now, but I don't believe they have a lot of currency.
But that is not the final word...you could assume a rapid change in the very early universe that approached the current value asymptotically such that $\alpha(t) \approx \alpha(\inf)$ for all time we can examine. This allows you to have funny things happening in the very early universe. However, in the absence of some failing of the conventional picture this seems pretty speculative.
As the observational evidence gets better and better, there is increasingly little room for such speculation, but only time (and a lot of effort, of course, but by people other than me...) will tell.
Many very accurate measurements seem to imply that the fine-structure constant has been totally constant, within the error margin, since the Big Bang. And the few experiments that suggested otherwise have problems. Theoretically, it's likely that the fine-structure constant is completely constant.
However, if you imagine that the fine-structure constant changes a bit, many things become different. For example, the stability and lifetimes of the neutron and nuclei will change abruptly. The modified fine-structure constant will also influence many other "constants" that define particle physics - such as the ratios of particle masses, fine-structure constants for other forces, mixing angles, and others - but you would have to describe what (dimensionless parameters) you want to keep fixed while you're changing the fine-structure constant.
From an "anthropic" viewpoint, it is important to notice that chemistry (minus nuclear physics, the care about isotopes etc.) would remain largely unaffected. It's because the atoms exist in the non-relativistic limit, and the fine-structure constant only determines the (order-of-magnitude estimate of the) speed of electrons in the atoms. Because the fine-structure constant is much smaller than one, the electrons in atoms may be approximated by non-relativistic mechanics. That would still be true if you e.g. doubled the fine-structure constant. So the ratios of the atomic frequencies etc. would remain pretty much unchanged, even if you doubled alpha, up to small corrections and splittings of the spectral lines (which are called the fine structure for a good reason).
It is even plausible that as complicated molecules as the DNA could continue to work unaffected even if alpha were changed by many percent or dozens of percent, despite the fact that the functioning of the DNA depends on small detailed in energy differences. If the units were redefined properly, the world of biochemistry would be pretty much unchanged. However, that's because the quantum electrodynamics with electrons only depends on one dimensionful parameter only - the electron mass - and one dimensionless one - the fine-structure constant that moreover defines a different scaling for space and time in the non-relativistic limit. If you considered the behavior of high-energy physics which has many other elementary particles, a change of alpha would surely make a difference.
Best wishes Lubos
This is a purely speculative idea, but it seems to me that the fine-structure "constant" is actually a measure of the compressibility of the vacuum. In a theory of quantum gravity, the structure of the vacuum (on which quantum fields are defined) and that of the background geometry are necessarily interrelated, in the sense that the vacuum can be thought of as a condensate of quantum geometrical excitations.
If the large-scale geometry of the universe can indeed be given a description in terms of such a condensate then the material properties of the condensate would show fluctuations over long distances. One of these would be the "vacuum" compressibility or $\alpha$.
Caveat Emptor: There are a few key notions that must be elaborated on before the above argument can begin to stand on its own. These are:
The first proposition is one I firmly believe in, not least because of what what I have learned through my own research. The second is something I am hazy on at present, though I have a strong intuition that such a description exists and is even natural.
I realize, that in the absence of a clearer explanation of these two issues I am asking the reader to believe in my reasoning based more on a "gut feeling" than anything else. This might be too much to digest for some. However, if for whatever reason, you are willing to give some credit to this line of reasoning then a cosmological variation in $\alpha$ is all but inevitable.