What are some circumstances, if any, that should make it a good idea to lynch on D2 rather than holding off till when you have more information?
Some people (i.e. me) would argue that it is always a good idea to vote to eliminate a player every day, because a day that you're not using that power is a day that the Evil team(s) get a free ride. The Village's job is to eliminate Evil and so I would argue you need a good reason NOT to vote someone on D2.
Many people would argue that giving the various intel roles more time to gather that intel without accidentally exposing them is a good enough reason. However, if there is no pressure on anyone on D2, useful targets to use that intel on are also more difficult to identify.
In a game with an open role-set, it is easier to make the judgement call, e.g. if there is a Bloodthirster, the Village would normally want to vote someone out in an attempt to reduce the KP the Wolves have. If there are Coven and/or Undead and no Wolves, the pressure is reduced, because the KP of those teams is lesser than that of the Wolves.
In a closed role-set, this can be much more difficult to identify, because you're unsure what Evil(s) you may be facing, but equally unsure as to the intel that you are waiting for, as there could easily be a Village team with no intel roles against a very small number of Wolves and you would be fruitlessly giving up valuable time.
This answer is definitely not coming from the usual meta of the site, but is coming from someone who would, if I could, enforce the Village to kill someone every day, including D1 if that were possible.
You do risk mis-killing members of the Village team, but I've always been one to subscribe to the "you can't make an omelette without cracking eggs" school of werewolfing.
Always. The only possible exception is when there are exactly four remaining players, and even that's debatable.
The best source of information in Werewolves is voting and lynching- people are forced to state opinions about a player, whose role is then revealed. Certain players are privy to knowledge about certain other players- werewolves- potentially revealing information.
If no lynch is voted for, you don't get this information. The werewolves get two night phases in a row, key roles get a chance to be killed off, and unless someone useful survives, gets information, and is willing to claim to reveal it, you put yourself in the same situation.