Make sure all external devices have been disconnected.
If you press the power button do you hear the startup chime? After a few minutes press the Caps Lock key to see if it lights up which indicates the laptop is powered on. It is possible you don't have a backlight or video to the display.
If you still have the problem (Caps Lock does not light up and no startup chime), then most likely the battery may be preventing the laptop from powering on. This usually happens if the battery has been completely discharged and sometimes if the battery has a problem. You can try disconnecting the battery from the Logic Board, then connecting the charger to see if the laptop will power on. If the laptop powers on with the battery disconnected, then try reconnecting the battery while the laptop is still powered on & booted. Let the battery charge at least 10% before disconnecting the charger or powering off the laptop, otherwise the battery may be permanently damaged. You just need to be careful not to let any metallic or conductive object or surface touch any of the internals, or you will damage the laptop. Placing the laptop on its right side or sitting the laptop on the display while open with the keyboard/palmrest in the air are the easiest ways to do this. You do not want to use any metallic or conductive tools inside the laptop while the battery and charger are connected. The battery connector can sometimes be a bit difficult (tight) to disconnect. You also do not want to puncture the battery or toxic chemicals will be released and possibly the battery will catch fire.
A 2014 MBPro is now considered "Vintage" so it is unlikely an Apple Store will repair it now. An Apple Authorized Service Provider can still repair this laptop if the necessary parts are still available and the AASP actually performs the repairs within their own shop. Only you can decide if it is even worth repairing a Mac this old which has at most two more years of macOS security updates and possible hardware support through an AASP.