No. For a limnic eruption to occur, the lake must have large amounts of CO₂ gas dissolved in it. First of all you need a source of CO₂, typically volcanic gases leaking out under the lake. Since Lake Michigan is not in an area with volcanic activity, it is not at risk.

Furthermore, the CO₂ gas has to stay dissolved in the water until it builds up to high enough levels to cause a limnic eruption. You can dissolve more CO₂ in water at high pressure and at cold temperatures. (For example, the CO₂ dissolved in a can of soda forms bubbles when you open it because it is now at lower pressure and can't hold as much dissolved CO₂.) This means that you get the most CO₂ dissolved in deep cold water. If that deep water comes to the surface, where the pressure is too low for it to stay in solution, then the CO₂ gas will escape into the atmosphere. This is what happens in Lake Michigan. Every spring and fall changing air temperatures cause the difference between the temperature (and therefore density) at the surface of the lake, and at the bottom of the lake, to be very small. As a result, the wind is able to mix the lake water from bottom to top, and CO₂ does not have a chance to build up very much.