In short, apparently not, but It may depend on the tournament. For example, the rules for both the FIDE Grand Prix and the next Candidates Tournament indicate that "At the end of each <standard> (i.e. Classic time control) game the players' original scoresheets are given to the Chief Arbiter." They don't say anything about Rapid and Blitz games so, presumably, the players are not required to record their moves for games played at the faster time controls. This is more and more the trend.

Yet for the FIDE Grand Swiss the tournament rules indicate that "At the end of each <game> the players' original score sheets are to be given to the Chief Arbiter." Of course, this probably has a lot to do with the Grand Swiss did not use Rapid or Blitz games as tie breakers. :-)

The FIDE Rules of Chess are different still. They indicate that "8.1 In the course of play each player is required to record his own moves and those of his opponent in the correct manner, move after move, as clearly and legibly as possible, in the algebraic notation (See Appendix C), on the scoresheet prescribed for the competition" and "8.4 If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then for the remainder of the period he is not obliged to meet the requirements of Article 8.1."

And when discussing Rapid play (after defining what that means), the FIDE Rules of Chess indicate that "A.2 Players do not need to record the moves." When discussing Blitz play (after defining what that means) they do not explicitly reference recording of the moves on the spreadsheet, but they indicate that "B.2 Where there is adequate supervision of play, (one arbiter for one game) the Competition Rules and Appendix A.2 shall apply.", with two exceptions that don't have anything to do with the recording of moves.