Heard many reports of people corrupting their OS by letting the phone die due to the battery running flat. If it doesn't get past the Samsung logo, then it could be the OS got corrupted.
The only two options I can think of is go into recovery and delete/wipe the cache partition, then restart the phone. Hopefully, the phone will start up. If that doesn't work, then you may need to factory reset the phone. You will lose all data, pics, contacts, etc. Hopefully you have all that stuff backed up to your computer or the cloud.
If you're lucky and the phone starts after wiping the cache partition, do yourself a favor and back your stuff up, and don't let the battery drain so low again... recharge around 35 - 40% and you'll never run into this kind of trouble (normally!).