What rule? If you're looking for an automatic draw that's not the way the rule is written.
code of chess

Yeah, you have to claim a draw for repetition or the 50 move rule. The only automatic draws are stalemates and draws on time when the person who still has time doesn't have enough material for checkmate.

Isn't there a rule in OTB where you can claim a draw by insufficient winning chances, for example if you were up a queen in a queen ending, or the above position? It seems like chess.com could try something like this.

If there's a lone pawn on the board, there's sufficient mating material. Although if there are equal numbers of pawns on each side butted up against one another, then none of them have the potential to promote . . .
EDIT: I should clarify, if the only other pieces on the board are Kings that can't approach the opponent's pawns from behind, then pawns butted up against one another can't have any chance of promoting . . .
I have done over 100 similar moves (in game 200!!!!!) without any captured piceses aginst my oponent, in the end of the game he had 1 pawn more and no chance to win (draw inevitable) and I lost the game on time :D as the computer chess live doest know the rule