Will my parachute fail if I'm going too fast when I deploy it?

TLDR: Yes, you will lose chutes from version 1.0.1

Kerbal Space Program has an advanced physics engine and while the planets of the Kerbol system are different to the Sol system, the mechanics are similar:

In real life a re-entering spacecraft is hitting air at huge speeds and that is heating the vessel. The chutes undergo huge stresses when deployed - both from aerodynamic pressure and heating. If these stresses are too great, the parachutes could simply tear off.

Version 1.0.1 of KSP brings this real life danger into the Kerbal universe. Now, if you're moving too fast when the chutes deploy, they can be "burned up". Or, even if you're not undergoing heating, the chutes can be destroyed due to aerodynamic pressures.

How can I avoid chute loss?

Not losing parachutes from heating is easy: Don't deploy them when your vessel is burning up. You can tell if the vessel is burning up by looking at it - is it on fire? Don't deploy chutes.

Not losing parachutes from aerodynamic stresses is also easy: Don't deploy (even partially) when you're going fast. How fast is "fast"? I've conducted some tests and it looks like 1000m/s is the magic number. Although it may change depending on atmospheric conditions.

Edit: After version 1.1 chutes can be destroyed going slower than 1000m/s. Look at the chute icon in the staging, it changes colour based on whether it's safe to deploy or not.

The rule of meassure I use is to only deploy the parachute when below the speed of sound (330 m/sec). I tend to wait until 250 m/sec, but I've never had a parachtute ripped off when under 330 m/sec.

The parachute doesn't do much anyway until you hit 500-1500m (Kerbin) altitude (most videos on Youtube show 500m, I'm playing 0.19.1 and it's usually roughly 1350m for me.). If at that point you're still going faster than 330 m/sec; may Kerbal God have mercy on your poor Kerbals' souls...

Edit: I do not use MechJeb, so I have no idea of its weight. The figures above are for a pod (doesn't matter how many Kerbals) and a parachute. If it's heavier, I'd scale those numbers accordingly.