The Geneva Convention treaties (1864, 1906, 1929, 1949) relate to the treatment of people in war - not the use of particular weapons. Specifically, they deal with humane treatment of wounded, sick, prisoners, and civilians. This does not extend to the execution of the war itself.
The use of particular weapons in warfare is related to the Hague Conventions (1899, 1907). Interestingly, the 1899 treaty bound the signatories to "abstain from the use of projectiles the object of which is the diffusion of asphyxiating or deleterious gases" and the dropping of projectiles from "from balloons or by other new methods of a similar nature." The bombing clause was only valid for 5 years. The gas clause was easily skirted. Thus, we see those weapons used in WWI. The Geneva Protocol to the Hague Conventions of 1928 is the one that bans the use of all forms of chemical and biological weapons.
So, no legal (or binding) prohibition against using elephant guns, shotguns, or saw back bayonets in WWI. Of course, all of those were minor inconveniences compared to the machine gun...
Related, but separate, the 1999 Ottawa Treaty bans the use of anti-personnel mines.
I'm sure this is too much information and going... But my diplomatic background got the better of me...