Magnesium Bicarbonate Water Reaction With MSM - Methylsulfonylmethane
On the reactions happening

It helps to have clear view of what actually happens at what point of the experiment. Before you add anything to your mineral water, $\ce{CO2}$ and water are in the following equilibrium (I am explicitly ignoring all other ions present, even if they are not spectator ions because they too can take part in acid-base equilibria):

$$\ce{CO2 (g) + H2O &lt;=&gt; CO2 (aq) + H2O &lt;=&gt; H2CO3 &lt;=&gt;\\ &lt;=&gt; HCO3- + H+ &lt;=&gt; CO3^2- + 2 H+}\tag{1}$$
$$\ce{CO2 (g) + 3 H2O &lt;=&gt; CO2 (aq) + H2O &lt;=&gt; H2CO3 + 2 H2O &lt;=&gt; \\ &lt;=&gt; HCO3- + H3O+ &lt;=&gt; CO3^2- + 2 H3O+}\tag{1'}$$

(Equation (1') is merely an equvalen version of equation (1) with extra water molecules to show that naked $\ce{H+}$ does not exist in solution.)

Under equilibrium conditions, most of the $\ce{CO2}$ is either dissolved ($\ce{(aq)}$) or present as minute gas bubbles ($\ce{(g)}$) and the amount of carbonic acid is rather low. Adding $\ce{Mg(OH)2}$, an Arrhenius and Brønsted base, shifts these equilibria rightwards by consuming protons:

$$\ce{Mg(OH)2 (s) + 2 H+ &lt;=&gt; Mg^2+ (aq) + 2 OH- (aq) + 2 H+ &lt;=&gt;&gt; Mg^2+ (aq) + 2 H2O}\tag{2}$$

Since reaction (2) will actively push equilibrium (1) rightwards, gaseous $\ce{CO2}$ is dissolved and then reacted, lowering the bottle’s overall pressure. Now you are adding methylsulphonylmethane. This compound can act as an Arrhenius and Brønsted acid by the following equation:

$$\ce{H3C-SO2-CH3 &lt;=&gt; H3C-SO2-CH2- + H+}\tag{3}$$

Hence, it is able to shift equilibrium (1) back leftwards, counteracting the force of reaction (2). The acidity of dimethyl sulphone (as it is also called) is due to the strong $-I$ effect of the sulphone group, which has a formal $\ce{S^2+}$ atom in its centre. The resulting anion can be understood in a similar way as phosphorus ylides can.

Since equation (3) pushes the equilibrium (1) leftwards, it again becomes more favourable to release $\ce{CO2}$ into the surroundings, i.e. form gaseous $\ce{CO2}$, i.e. increase the bottle’s pressure.



On the safety aspects

You should never drink anything that has come into contact with laboratory grade chemicals. These are not tested to be safe for consumption, they are tested and controlled only to be safe for use in a lab (and hence in chemical reactions). There is at least one urban legend of a fatal accident happening due to mixed-up chemical bottles.

That taken care of, it seems that dimethyl sulphone be a relatively safe additive. According to Wikipedia, its $\mathrm{LD_{50}}$ value is larger than $17.5~\mathrm{g / kg}\text{ body mass}$, making it a safe chemical. It is sold as a dietary supplement, showing that the risk seems to be very low. As such, it will probably be safe to drink the soda after addition of both compounds, provided the pH value is where it should be (i.e. dimethyl sulphone has been added sufficiently to counteract the pH increase induced by $\ce{Mg(OH)2}$.)