There are a few examples in computer games that have no combat, often focusing on avoidance of combat entirely for example amnesia - dark descent. There is a great game called the sexy brutale (I mentioned this elsewhere as well. It really is great). It is a series of murders that happen on the same day set in a casino / masquerade ball. It's groundhog day style so you can play through the same day numerous times in an effort to prevent the deaths. You cannot interact at all with the NPC's. Instead you must just observe and keep track of the movements and actions of the NPC's on that day and investigate rooms whilst they are not there.
For tabletop RPGs I know one example: mansions of madness. There is combat, however it is generally a bad idea to do it because each scenario is quite a challenge to get through with your health and sanity intact so you are better off avoiding combat.
The focus in that game is the investigation of the mystery in each given "mansion", and the unravelling of a broader plot. This is achieved by each (of up to four) investigators having 2 actions which they can spend either moving, or collecting clue / item tokens on the map. When you encounter NPC's you can also decide what to do with them: speak, observe their actions etc which will also give you clues. At certain points events are triggered as well which, usually, force the party to hurry up by introducting quite a pressing challenge. If you don't get to the bottom of the mystery in time then you all become too injured or driven insane and so fail that scenario.
There is a heavy lovecraftian theme. So for example so you may get presented with a mystery around a cult, and you must discover who the leader is and kill them. As you reach the midpoint dark creatures end up infesting each room making it increasingly difficult to complete your investigation. Another one is that someone in a stately home is possessed and you must discover who in a certain time frame to prevent a portal being opened.