There are a couple of common myths (there are more, but most of them are just variations of the two) surrounding Lithium Ion and Lithium Polymer batteries, most of them coming from the fact that people don't know the difference between the two, and from people mistakenly thinking that what applies to other rechargeable batteries also applies to Lithium-based ones.
Myth #1 - Charging the Battery When It's Not Fully Drained Will Shorten Its Life
This is called the "memory effect", wherein the battery gradually loses maximum energy capacity when regularly recharged after only being partially discharged.
The Truth - this is an example of a myth stemming from people thinking that Lithium based batteries behave the same as other kinds of rechargeable batteries. The memory effect only applies to Nickel Cadmium batteries (Ni-Cad), and actually happens on such a slow scale that it will take a hundred or so discharge/recharge cycles before the effect is noticed.
Li-Ion and Li-Poly batteries, on the other hand, are immune to the "memory effect", and in fact will benefit from being recharged after a partial discharge. This is because Lithium based batteries have a set number of discharge cycles, after which they will start to lose charge capacity. So if you use it up to 50% capacity, then charge it to full capacity, you have only consumed half of a discharge cycle. Using it up to 50% capacity then charging it to full again reduces the charge cycle by another 50%, and so on.
Myth #2: Overcharging Is Dangerous and Will Make Your Battery Explode
This is not a full-blown myth, since there is some truth behind it. Lithium-Polymer Batteries CAN explode when overcharged. However, they all have special circuits that either halt or reduce the charge rate when full. This means a Li-polymer battery will only overcharge if it's defective, therefore, you'd need to have a defective battery before it explodes.
You don't have to be afraid of leaving a battery plugged in even when it's full, like when you've just charged a laptop to 100% and decided to keep on using the AC power because your laptop doesn't let you use its discrete graphics card when using battery power. However, you should still be sensible about it. Leaving a full battery plugged in and charging when you're not using it for long periods of time is just bad habit, and a sign that you're not taking good care of your gadgets.
The problem with the two myths above is they result in people actually doing harm to their batteries, such as continuously draining their batteries before recharging ensuring that they go though the set number of discharges faster than normal. Lithium-based rechargeable batteries are not new inventions by any means, but they are a big improvements over old nickel-based rechargeables, both in capacity and ease of use. Don't waste the benefits they provide just because of myths that have been passed down by people who don't know any better.