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Only group III & IV are used for doping in elemental semiconductors like Si and Ge, why can't other groups be used instead?

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Several points to add to @SamuelWeir's answers. Here I will focus on data for silicon, but the general principles apply to other semiconductors.

First, plots of dopant energy levels are fairly common in semiconductor physics books. My copy of Sze, Physics of Semiconductor Devices, has it as Chapter 1, Fig. 13 on page 21. For silicon, the shallowest donors are Lithium (0.033eV), Sb (0.039eV), P (0.045eV), As (0.054eV) and Bi (0.069eV). Mg (0.11eV) and Ta (0.14eV) are somewhat surprising entries. (Note that Mg and Ta are amphoteric - they have multiple levels within the gap and the ones listed are the shallowest. This will greatly limit their utility as donors.) On the acceptor side, one has B (0.045eV), Al (0.067eV), Ga (0.072eV) and In (0.16eV).

Now, besides having a useful level in the gap, you actually have to be able to incorporate the dopant in the lattice. So, the solubility of the elements in crystal silicon is important. The classic reference for solubility is F.A. Trumbore, "Solid Solubilities of Impurity Elements in Germanium and Silicon", Bell System Technical Journal 39(1) 205-233 (1960). One quickly notices that B, As, and P all have solubilities near 10$^{21}$/cm$^{3}$, or better than 1 at.%. The next grouping falls below 10$^{20}$/cm$^{3}$. This group includes Sb, Ga, Al (on the low end) and Li. Bismuth has a solubility below 10$^{18}$/cm$^{3}$, which is useless for modern device technology. So the utility of B, P, and As is pretty clear - they are good dopants and you can get a lot in to the lattice.

Finally, a note on Li - it is a very shallow donor, and the solubility is as good as Ga and Sb which were commonly used in the past (not so much now). However, lithium diffuses quickly, and the diffusion is enhanced by electric fields. Both of these are, not surprisingly, bad for device performance long term. (Although it is quite useful to make very deep junctions for, e.g., particle detectors.)

So, the choice of dopants relies on a number of factors - levels in the gap and solubility are the main ones, but other factors can also come in to play.

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I believe that the reason may be at least two-fold: (1) For dopants to be effective, the energy level introduced by the dopant has to be a shallow energy level, not a deep energy level. A shallow level means that the impurity level is very close to the valence or conduction band, so it is easy to thermally ionize the dopant atom and have its electron (or hole) contribute to the electronic behavior. An impurity which only contributes a deep energy level is effectively inert as far as its contribution to the electronic behavior goes because the thermal energy is too low to ionize the impurity. Group III and Group V elements are similar enough to the Group IV elements silicon and germanium that they tend to contribute shallow impurity energy levels. Doping with other elements will provide impurity energy levels, but they will tend to be much deeper and, hence, effectively inert energy levels.

Another issue may be (2) chemical/physical compatibility. Silicon and germanium both crystallize in a rather dense diamond cubic structure, and so many possible impurity atoms would probably be too large to conveniently fit into the silicon or germanium lattice as a substitutional impurity the way that relatively small atoms such as, say, boron or phosphorous do.

By the way, synthetic diamond grown by CVD is being viewed as a possible electronics material of the future since its charge carriers (i.e., electrons and holes) move much faster than those in silicon, and since the thermal conductivity of diamond is very high, which means less likelihood of the device overheating. But a big problem has been finding suitable electronic dopants for diamond. Diamond has an extremely tight lattice, so most atoms will not fit into diamond as a substitutional impurity. Also, the possible dopant atoms studied thus far all tend to contribute rather deep impurity energy levels into diamond, not the shallow impurity energy levels that are desired.

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