If I were to fail a dexterity roll to unlock a chest, could I redo it next time? I know that if you fail a skill challenge it is permanent, but are skill checks the same, or can you redo them? If you could cite the page number describing this in the DMG or PH, I'd appreciate it.
It largely depends on the consequences of a check. If there isn't a consequence for failure that would stop or deter you from trying again, then there's no reason you wouldn't be able to (the permanent failure for a skill challenge is in that it still counts as one failure for purposes of the skill challenge, even if you try again and succeed next time)
Take for example if you were trying to cross a balance beam.
First attempt you roll poorly on your acrobatics and fall to the ground, maybe you hurt yourself, maybe the person chasing you gets closer, and maybe you just look like a fool. Now that you've failed though, does it make any sense that you would never be able to try to cross the balance beam again?
The same goes for the chest... to an extent. This, as with all checks, is up to the DM. You might not be able to try again for a variety of reasons. Perhaps if you fail by too much you jam the lock to such an extent that it cannot be picked, or even opened with the key! Failure could mean you make a bunch of noise in the attempt and alert a guard. Maybe you spring an unnoticed trap that turns the contents to marshmallow fluff... the options are limitless. Even mundane ones like saying that when you attempt again, unless you've done something to give yourself a new advantage, you're going to "roll" the same.
If there is no consequence for failure, then the question becomes why is the skill roll being made in the first place? If it is possible for someone to succeed and there is no limiting factor, then in theory they can try until they succeed. In such situations the roll serves no purpose to the game or to the enjoyment of anyone involved and whether they succeed or fail should simply be narrated (possibly with ease of success being decided by a single attempted roll). Of course if it's something someone could only get on a 20 then trying until they get that 20 could take a lot of time in game, but this should only be rolled out in a situation where that time spent matters, and is thus a consequence for failure.
So the real answer is, ask your DM what happens if you try again, and check to see how much they grin evilly.
If you are the DM in this scenario, than regardless of what you have planned or they roll, grin evilly.
Note As mentioned in BESW's comment, the PHB explicitly allows "taking 10" in non stressful situations (PHB 179). Previous editions allowed taking 20 when you had all the time in the world to do an action, it just took 20x as long as attempting to do the action once and any possible things triggered by failure would be triggered during the attempt.