Should you let the minions destroy the towers?

This is done to deny your opponents farm. If you can push your opponent out of the lane and let their tower take the minion kills, then they are missing out on valuable experience and gold.

This also prevents your lane from pushing too far, which might put you at a disadvantage where you have to farm too far away from your own base.

The tower is pushed to a low amount of HP so that, when the need arises to take the tower down later in the game, it's really quick to finish it off and move onto the next objective.

In lower Elo, this isn't actually as important. Your opponents were probably going to miss some of the last hits anyway, and they will likely push your lane back harder allowing you a safer farming position, or telegraph their gank, or not even gank you at all if you're in an extended position. It doesn't hurt to try to play optimally, but you probably won't get noticeable gains from doing this compared to in higher Elo.

StrixVarias Answer is Correct. You let the turret survive to deny creeps to the enemy. However it's important to notice that this is not appropriate in every situation and especially in soloqueue you should take it down more often. It's important to know when to kill or leave a turret.
Only the outer turrets should be left alive when you have the option to kill them

Mid

Most of the time when playing midlane you should focus on taking down the turret early since it oppens a better gank opportunity through the jungle for your teammates and it also allows you to roam a lot more. Also Most midlaners aren't too good at pushing turrets so it will take you a while to take one down and it can't hurt to just hit it whenever it's safe to do (most midlaners are either AP -> No AA damage or meele AD -> Risk of getting ganked). So basically you should rush turrets when playing midlane. There are some exceptions of course but around 80% of the time you can go for it.

Bot

For Bot it's not exactly the same. Usually It's a big advantage to take down the Blue side turret since it allows you to control Dragon a lot better. The purple Turret however isn't too important. If you stand behind the purple turret you're a lot more vulnerable to ganks. Also if you are playing a longrange or poke ADC you shouldn't rush a turret since you can even poke them with the turret still alive while the turret itself denies them Gold.

Top

Toplane is a bit tricky. There's no situation on where you could say "Hey kill the tower!" Or "Let it survive!". Another big problem is the fact that most toplaners run teleport nowadays. This makes taking down a turret harder especially early and If you are a meele champion (like most toplaners) you also become vulnerable to a jungler/midlane gank when pushing a turret. For soloqueue (especially low elo) it's often a good tactic to just stay top and push all day. Low elo players tend to ignore the fact that there is a Darius at their inner turret and they focus solely on teamfights. You can see this mistake even up to lower diamond (IV-III) but of course not that often. Also If you realize that the enemy doesn't ignore the splitpushing, you're better off helping your teammates instead of dying 2v1 all the time.

Oh and as a final note, since you mentioned it: Never let minions kill the turret alone

Make sure to always land at least one hit on the turret within 10 seconds of its death since you will gain some extra gold which would otherwise be lost (Of course only if you are near the turret).

The previously existing answers are now outdated. You should always kill a turret if you are able.

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NEW BUFF - PUSHING ADVANTAGE

If your team's average level is higher than the enemy's:

• Your minions deal 10% bonus damage to enemy minions. If your team has taken more turrets, this damage is increased by a further 10%.
• Your minions take 1 + (turret differential × level differential) less damage (flat) from enemy minions.

If your average level is at least 3 levels higher and you've taken 2 more turrets in a single lane, your minions become Enraged:

• The bonus damage is increased by a further 90%.
• The flat damage reduction is increased by a further 7.

Some emphasis added on relevant text

Killing a Turret also gives a significant amount of gold, which is important for you and your team, especially if it's the first tower to be taken, which gives 400 more.

Sources:

• http://leagueoflegends.wikia.com/wiki/Minion#Patch_history
• http://leagueoflegends.wikia.com/wiki/V5.23