There are two timeouts for your fire stick:

• Time until the screensaver starts
• Time until the screen goes dark

The first one can be set in the preferences menu, the second only via adb. Enable debugging support on you device and connect via adb.

adb connect IP_ADDRESS
connected to IP_ADDRESS:5555
adb shell

Change the timeout:

shell@montoya:/ $ settings put system screensaver_timeout 30000
shell@montoya:/ $ settings put system screen_off_timeout 214760000
shell@montoya:/ $ settings get system screen_off_timeout
214760000