The burn - or more commonly "pump" - is the result of microtears in the muscles you've worked. So in essence, yes, the harder and longer you work a muscle, the more it will 'burn', and the more you will be sore in the following days. This isn't necessarily a good thing, however; it only indicates that you've worked a muscle longer, not that it will grow bigger. If you're eating properly for tissue regeneration, then by all means, go for the burn...but chances are you'll only ever get "the burn" through volume training (which, to the lay person, is many sets of many repetitions {generally nauseating work, and not conducive to prime growth}).

In summation, the burn doesn't indicate growth, it indicates damage done to the muscle. The growth will take place immediately post workout and in the following days, depending on what you put in your body and what you do with your time.