On this page, I explain the draw by "impossibility of checkmate", and show examples where it is violated even by grandmasters at major tournaments.
The draw is stated three times in the FIDE Laws of Chess:
Article 1.3 If the position is such that neither player can possibly checkmate, the game is drawn. Article 5.2b The game is drawn when a position has arisen in which neither player can checkmate the opponent's king with any series of legal moves. The game is said to end in a 'dead position'. This immediately ends the game, provided that the move producing the position was legal. (See Article 9.6) Article 9.6 The game is drawn when a position is reached from which a checkmate cannot occur by any possible series of legal moves. This immediately ends the game, provided that the move producing this position was legal.
Let's look at the game Shakhriyar Mamedyarov (2757) vs Viswanathan Anand (2786), played in Dortmund 2007, round 5.
Position after 68.b6. This is a live position because there still exists a sequence of legal moves that would lead to a checkmate. For example, 68...Kc8 69.bxc7 Kb7 70.c8=Q+ Ka7 71.Bd4#. Position after 68...Kxb6. This is a dead position because neither player can possibly checkmate (note how the two bishops are on squares of the same color). This immediately ends the game, as explained in articles 5.2b and 9.6. Mamedyarov inexplicably plays 69.Kd4. This move is illegal because the game already ended, according to the rules of chess.
At this point you might expect Anand to refuse the illegal move, or to see the arbiter rush to the stage to correct the mistake. No such thing happened. What about a subsequent media uproar? None. Apparently no one noticed, except Noam Elkies who posted about it on a chess problem mailing list.
Although the draw by impossibility of checkmate rule is easy to state ("neither player can possibly checkmate"), it may help to think of all the ways this can happen.
To find more examples of illegal moves, I used my chess engine to test games from the ICOfY Base (version 2011.1) which is a free database of 4.2M+ chess games, and from some other sources. The draw by impossibility of checkmate rule went into effect on July 1st, 1997 (FIDE Laws 1996/1997, articles 1.3, 9.6), so I'm limiting my examples to year 1998 and later. Before that, the rules were different (FIDE Laws 1992/1993, article 10.4) and only positions with insufficient material immediately ended the game.
Below are all the games that I found from 1998-2011 where an illegal move was played by a 2700+ rated player (in bold). (Here's a PGN file with all the games if you prefer.)
Date Event site Game Illegal move Reason 1999-08-?? Las Vegas Michael Adams (2708) vs Aleksey Dreev (2679) 75.Ke4 IM 2004-01-27 Catalan Bay John Emms (2501) vs Nigel Short (2702) 94... Kd4 IM 2005-05-16 Sofia Viswanathan Anand (2785) vs Michael Adams (2737) 54.Kxb4 IM 2005-09-19 Saint Vincent Alexander Grischuk (2720) vs Boris Avrukh (2652) 65.hxg5 FS 2005-09-21 Saint Vincent Etienne Bacrot (2729) vs Viktor Bologan (2700) 139.Kxf8 IM 2007-06-29 Dortmund Shakhriyar Mamedyarov (2757) vs Viswanathan Anand (2786) 69.Kd4 IM 2007-09-28 Mexico City Alexander Grischuk (2726) vs Viswanathan Anand (2792) 74.Ke4 IM 2008-04-10 Dagomys Shakhriyar Mamedyarov (2752) vs Dmitry Andreikin (2580) 68.Ke5 IM 2008-05-09 Sofia Bu Xiangzhi (2708) vs Teimour Radjabov (2751) 45... Kxh5 IM 2008-08-08 Sochi Dmitry Jakovenko (2709) vs Sergey Karjakin (2727) 57... Kg5 IM 2008-08-27 Moscow Gata Kamsky (2723) vs Ruslan Ponomariov (2718) 74.Kxc7 IM 2008-10-03 Moscow Nikita Vitiugov (2638) vs Evgeny Alekseev (2715) 84... Kxd2 IM 2008-11-16 Dresden Vassily Ivanchuk (2786) vs Levon Aronian (2757) 68.Kxh7 IM 2009-01-17 Wijk aan Zee Loek Van Wely (2625) vs Leinier Dominguez (2717) 80... Kxh4 FIM 2009-03-03 Linares Leinier Dominguez (2717) vs Levon Aronian (2750) (*) 56... Kxg7 FS 2009-08-23 Sochi Vladimir Malakhov (2707) vs Ding Liren (2530) 84.Kxd5 IM 2009-12-01 Khanty-Mansiysk Ruslan Ponomariov (2739) vs Etienne Bacrot (2700) 93... Kxe4 IM 2010-05-16 Astrakhan Vassily Ivanchuk (2741) vs Teimour Radjabov (2740) 58.Kd4 IM 2010-06-04 Leon Francisco Vallejo Pons (2703) vs Boris Gelfand (2741) 67.Bxg5 IM 2011-01-16 Wijk aan Zee David Navara (2708) vs Le Quang Liem (2664) 58.Kxe5 IM
(*) Game obtained from PGN Mentor (Linares2009.pgn), somehow missing from the ICOfY database.
Such illegal moves are particularly common with 2700+ rated players because recent top tournaments often forbid draws by mutual agreement, which means that players are more likely to reach an unfamiliar draw by impossibility of checkmate.
Special thanks to Andrew Buchanan for teaching me this rule in 2004, and to Noam Elkies for pointing out its violation in professional chess.