Random resolution

 We all know a lot of good reasons to roll dice in D&D -- impartiality, inspiration, injecting the unexpected into a game. What I'm looking at in this post is when it's appropriate to use random rolls to resolve character actions in-game, whether as a formal game design choice or as an ad hoc ruling during play.

The first qualification: Assuming success and failure are both realistic possibilities, does failing at this action have interesting consequences? If not, there's no point in dragging things out. Just say "yes," maybe apply a cost in terms of time or party resources, and move on.

The second qualification is more complicated: Is it possible to fail without touching the dice, i.e. by a player's mistake or oversight?

Failure is an inherent risk of relying on player skill. A player can misjudge a risk, overlook a detail, or fail to realize the significance of a cue or clue. This happens all the time even (especially!) in games with no random elements at all, else nobody could ever win or lose a game of chess or checkers. 

In D&D, players can fail to find a secret door by not choosing to search where it is, or by failing to interact with features the DM describes in the right way to trigger it. Players can fail to detect a trap if they misinterpret or decide not to investigate the scorch marks, dried blood, or pulverized stone you describe in front of a door. Players can fail to acquire information by choosing to attack first and ask questions later, or by shrugging at that shelf full of moldy books in that out-of-the-way dungeon chamber. Players can botch a social encounter by not heeding cues, by acting unnecessarily belligerent, or many various other ways. Handled well by the DM, none of these situations need random determination, let alone character skill, to resolve them.

Picking a lock is another matter. There's no clear point at which player skill could fail. It's not normally something that is hidden, so the player isn't going to botch finding the lock. It's not a case in which most players could describe in detail how their character picks the lock, beyond the very obvious ("I insert the lockpick in the lock and poke around, keeping my hand very steady.") Unless you want lock-picking to be an automatic success (which pretty much negates the point of locks,) it's a very good candidate for a random resolution mechanic, perhaps modified by a character skill dedicated to that very task. 

Bottom line: If a possible failure state exists in player actions alone, it's usually more fun not to diminish the players' agency by bringing dice into the equation. If there's not a baked-in window for failure, and failure would drive interesting choices in play,  a random mechanic may be appropriate.