If the battery is completely drained, you could have to let it charge for about 15 mins before you see something on the screen.
Below is Apple's suggestion for this problem, if it doesn't fix it then you'll have to send it in for service.
iPhone: Hardware troubleshooting
Will not turn on, or will not turn on unless connected to power
Verify that the Sleep/Wake button functions.†† If it does not function, inspect it for signs of damage.†† If the button is damaged or is not functioning when pressed, seek service.
Check††† if the Liquid Contact Indicator (LCI) in the headset jack or dock connector is activated or there are signs of corrosion or debris. Learn more about LCIs.
Connect the iPhone to the iPhone's USB power adapter and let it charge for at least ten minutes.
After at least 15 minutes, if:
The home screen appears: The iPhone should be working. Continue charging it until it is completely charged and you see this battery icon in the upper-right corner of the screen . Then unplug the phone from power. If it immediately turns off, seek service.
The low-battery image appears, even after the phone has charged for at least 20 minutes: See†† "iPhone displays the low-battery image and is unresponsive" symptom in this article.
If something other than the Home screen or Low Battery screen appears on the display, continue with this article for further troubleshooting steps.
If the iPhone did not turn on, try resetting it while connected to the iPhone USB power adapter. To reset, press and hold the Sleep/Wake button and the Home button at the same time for at least ten seconds, until either the Apple logo or the battery symbol appears.
If the display turns on, go to step 4.
If the display remains black, go to next step.
Restore the iPhone. Connect the iPhone to a computer and open iTunes. If iTunes recognizes the iPhone and indicates that it is in recovery mode, attempt to restore the iPhone. If the iPhone doesn't appear in iTunes or if you have difficulties in restoring the iPhone, see this article for further assistance.
If restoring the iPhone resolved the issue, go to step 4. If restoring the iPhone did not solve the issue, seek service.