You're dealing with two considerations here. The mower is rated for 12 amps, but probably uses less when running at normal speed and cutting moderate grass. A 16 or 14 gauge cord will handle that with minimal heating of the cord itself. And no, it cannot cause excess heating or overloads inside your house, leading to a house fire.
The problem really is in what happens to the mower motor under these steady-state conditions, and under sudden load conditions. Because of the current being drawn, there is a small voltage drop along the 100' electric cord, so the motor is not getting all the voltage it needs. So it will pull slightly more current and generate more heat than normal, thus running a bit warmer than design. But what really hits hard is when you first start it up - that's when the starting current is MUCH larger (but only for a few seconds). The combination of low voltage requiring more current, and undersized cord causing even more voltage drop, makes this starting current effect larger than the designers intended. Likewise, when you hit heavy grass and the motor needs to pull more current to keep cutting, the same effect overloads and overheats the mower motor because of voltage drop. The worst case is when you start it up already sitting in heavy grass so the motor is trying to start under heavy-load conditions.
All of these things are much lees of a problem if you use a 100' cord designed for MORE than 12 amps. Several have recomended 12 gauge, which is good for over 15 amps usually. I agree. Even under heavy load conditions that cord will produce much less voltage drop.
The alternative of stringing together two 50' cords rated for 13 to 15 amps gives you the current-carrying capacity (and hence lower voltage drop) you want, but it has two problems. One is mentioned by a few here - the possibility of electrical leakage at the junction point when it is wet. To me there's another equally important problem. Any junction - a plug into a cord-end socket - is a place with imperfect contact and hence resistance. That resistance, causing voltage drop and heat generation at the contact point, gives you just as much trouble as an undersized long cord, maybe even more. And it will only get worse as the cords age and the contact point oxidize.