If your game needs to run on a network, there is nothing you can really do, as synchronization with other clients is the top priority—you simply can’t allow the game to decrease its logical update speed etc.
If it is a single-player game, you have the option to decrease the speed of the game so that it all appears to slow down for the player, probably the lesser of 2 evils.
This is easily implemented by adding a cap to the number of logical updates that can take place in a single frame, and when the maximum is reached, consider any logical updates you had to drop as a loss and do not try to recover them on the next frame.
To answer your actual question, the side-effect is that your game will eventually become unresponsive and considered dead by the player.
L. Spiro