Native French speaker and regular tournament-goer (pre-Covid) here. The comments under the other answer are correct.
Some background tournament etiquette (probably not France-specific but necessary for what follows):
- Calling the arbiter: stop the clock, raise your hand and say "arbitre" (once, but loud enough to be heard). Do not leave your place.
- (pre-Covid, this may have changed) You shake hands with your opponent before and after the game.
Now for the French sentences:
- Offering a draw: see Evargalo's comment, "je propose (la) nulle" is IMO best; just "nulle ?" sounds quite rough to me but it is not offensive either. Refusing such as offer: "non" (no); accepting: "oui" (yes); or any other non-verbal variant (for instance, extending the hand over the board after such an offer = let's shake on it = yes).
- Claiming a draw by repetition: "nulle" (as an affirmation, not a question). Should the opponent disagree (I have never seen that happen), call the arbiter: "la même position a été répétée trois fois" (the same position has been repeated three times). "Je réclame la nulle par répétition" (I claim a draw by repetition) is correct, but formal.
- Illegal move: if you feel nice, you can point out that the move is illegal and allow the opponent to retract it ("ce coup est illégal", this move is illegal, or something more precise, for instance "on ne peut pas laisser le roi en échec", you cannot leave the king in check). The officially-correct thing to do is to call the arbiter who will apply whatever the rules say (if memory serves, in longer time controls, the opponent gets extra time for the first two offenses).
- Resigning: "j'abandonne" (I resign). Common non-verbal alternatives: tipping the king over (which some people consider mildly rude, but IMO it is not if you do it calmly and with a smile), or extending the hand over the chessboard (to shake it, implying the game is over).
- Reminding someone of the touch-move rule: "pièce touchée, pièce à jouer" (a touched piece must be played). Less useful: "pièce lâchée, pièce jouée" (a dropped piece has been played = you cannot change the square you put it on). (Only for intentional contact - if the opponent reaches for their glass across the board and accidentally touches a piece, it does not apply.)
- Adjusting pieces: "j'adoube" (must be said before touching a piece)
- Pairings: "appareillement(s)"
- Leaderboard: "classement"
Game notation is algebraic, with the following abbreviations: R=roi (king), D=dame (queen, literally "lady"), T=tour (rook, lit. "tower"), F=fou (bishop, lit. "fool"), C=cavalier (knight, lit. "rider"). For instance Alekhine's defense is written 1.e4 Cf6 instead of 1.e4 Nf6.