A lot of people have encountered issues with the Note 3 battery draining rapidly and unexpectedly. You should check out the steps for the overheating problem below, as this is likely caused by the same things. You can also check out our smartphone battery saving tips for general advice. But there is another potential fix.
- Take a look in Settings > General > Battery and see if you can find anything obvious causing a problem. A faster way is to just backup anything important and do a factory reset. A lot of people report improved battery life after this. Go to Settings > General > Backup and reset > Factory data reset.