There are a lot of issues that can strike your plant. The issues and solutions are easy to list but the symptoms are pretty extensive and many times the same symptom could be one of several issues. The best solution is whenever you see something you're concerned about just take a good clear photo and post it here.
With that said, here are some things to look out for:
Too much water: You likely won't see the symptoms until rot sets in, when it does you will see leaves going limp even though the soil is plenty moist. If the rhizome starts to rot you run a good chance of losing the plant. Solution: Water less! Keep it moist.
Too little water: Leaves will start to look dry and less plump. Usually comes from oversight, over watering is much more common. Solution: Water more, keep it moist.
Sun burn: Just turning red is normal and not a problem. Actual sunburn is seen as darkened/ dying areas on leaves. Seen either near the base in the case of a plant not planted deep enough or randomly all over in the case of a plant not acclimated properly. Unless the plant is very small it rarely results in death, just an ugly plant until new leaves grow in. Solution: Be sure to know how the plant you received was grown and acclimate it accordingly.
Insect / Animal damage: This one is obvious. Leaves may be partially eaten or all of them are chewed off, or they have small dots of damage from sucking insects. With a few exceptions(mites, aphids) this usually doesn't kill the plant unless it's really small. Solution: Depends on the specifics. Mites, use a miticide. General insects use an insecticide like acephate. Birds, deer, cats, etc., find a way to keep them away from your plants.
Fungus: Random dying spots not attributed to sun burn. Root rot is a type of fungal infection, though a serious one. For soil level and below use something like daconil spray. For leaf based infections you can use a sulfur based powder or spray.
Mineral burn: Consistent malformed leaves. Small undeveloped traps. Generally a bad looking plant. If caught early you will likely not lose the plant. Solution: Un-pot and rinse the plant in rain/RO/DI water. Re-pot in an approved CP soil mixture. Also, don't fertilize your fly traps.
Not enough sun: Long skinny floppy leaves. Sometimes shorter wide leaves with small traps. It can really vary depending on how lacking the light is. You need to know what cultivar you have and know what it should look like. Some always grow long spindly leaves no matter what. Solution: Duh, give it more light.
Some things that may look like issues but are normal:
Trap death due to age: Trap starts turning brown at the edges. Or it just won't close anymore or starts turning outwards. You should still be seeing new healthy growth coming from the growth point. Solution: Don't worry so much, it happens.
Trap death due to indigestion: Trap closed on a big bug or couldn't seal all the way. Starts turning brown just around the trapped bug. Solution: Don't worry so much, it happens.