If you think your opponent has missed a draw sometimes checking the river when the highest card pairs can be a smart play from OOP.
A few weeks ago I found myself out of position to a maniac. This guy came to the casino to gamble and did not care if he called a raise with 73s or three bet you preflop with J5os. He also had the tendency to bluff rivers when he missed his draws even in spots where it was obvious that he was going to get called.
In this particular hand he had been winning a little bit and raised from MP2 to $40. I was in the big blind with 88, $1000 effective and called. The board came out K♣ 7♣ 6♥. I checked to him and he made a $60 continuation bet. I called, thinking I could possibly have the best hand. The turn was the 8♥, bringing a backdoor flush draw and giving me a set. Not wanting him to check through on the turn I decided to come out and bet $130, which he quickly called. At this point I thought that it was likely that he could have a draw because the board was so wet or a king. The river brought out the K♠ and I thought for a moment about the most optimal play. Not only did I think that he would bet trip kings at the end if checked to but I thought if I checked the king that that would give him the green light to bluff at his draw—since so many people would not check trips to him from out of position in this situation especially if they took a check-call lead turn line. So by not betting I induce him to bluff the king and I also do not lose value if he has a king since he will do the betting for me.
Even though this opponent was extremely wild you can absolutely use this concept against more standard types of opponents. When the top card pairs in a limped pot or you have taken away the betting lead from the preflop raiser often times by checking at the end you show a lot of weakness. The one exception to this is when you are the preflop raiser and have shown a lot of strength. In that case checking the river when the top card pairs makes it look like you have an overpair and you are playing it cautiously but in actuality are ready to check call.