It is my second main phase and I forgot if I played a land this turn. If I ask my opponent, is he required to tell me the right answer, or if he does not know, at least to cooperate with me while we try to figure it out? Or can he simply wait until I play another land and then call a judge if it was incorrect?

Can I ask a judge, “please help us determine if I played a land this turn” before an infraction occurs?

Basically, the question is: “Is the fact that a land was played this turn free information?”

“Did I play a land this turn?” Is a question that is associated with details of past game actions that affect the game state. The opponent, and even judges, are encouraged to help the player determine the answer.

As Eric said, it's about free information. MTR tell us: “Players must answer completely and honestly any specific questions pertaining to free information.”

So, they are encourage to help and they're obligated to respond completely and honestly. Silence is not an option.

Related question: What if the player instead turns to me, a spectator, and asks that question? Am I obligated to do anything in that case? (I had this happen to me as a spectator when watching a match at a GPT and the player forgot whether he'd used Jace, the Mind Sculptor that turn. As I recall, I replied, “I'm not saying anything,” because I was afraid of running afoul of an OA infraction.)

Outside Assistance specifically applies to play advice or hidden information, so that would not apply for this adjusted scenario of if the player activated her Planeswalker this turn. As that information is free information available to both players, similar to having played a land this turn.

Originally posted by Infraction Procedure Guide: Outside Assistance:

A player, spectator, or other tournament participant does any of the following:
Seeks play advice or hidden information about his or her match from others once he or she has sat for his or
her match.
Gives play advice or reveals hidden information to players who have sat for their match.
During a game, refers to notes (other than Oracle™ pages) made before the official beginning of the current
match.

There is however the question of perspective and how it would appear if a spectator did respond to such a question and that is a murkier issue. You are better off pointing the player either to her opponent or better yet a judge to assist them in making such a determination if there is a questions, especially at competitive REL. Though this last part is just my two cents.

Originally posted by Daniel Pareja:

Related question: What if the player instead turns to me, a spectator, and asks that question? Am I obligated to do anything in that case? (I had this happen to me as a spectator when watching a match at a GPT and the player forgot whether he'd used Jace, the Mind Sculptor that turn. As I recall, I replied, “I'm not saying anything,” because I was afraid of running afoul of an OA infraction.)

A spectator is not to assist players during a match, with the exception of going to obtain a Judge. That said, they are more than welcome to assist a Judge with what they saw and know to help players in that regard. Beyond that, the rules of public/derived/private information are specifically meant toward the players in that game.