Make sure you are playing as Ethan
This is the solution of the birthday puzzle when you play as Ethan and not the one on the video tape. If you want the solution of the puzzle on the video tape, check this link.
Grab the candle
Grab the lit candle from the hands of the dummy statue.
Remove the wooden part
Go to the room with the birthday cake and find the wall blocked by wooden parts. Remove a wooden part to create a hole in the wall. You will need that hole later.
Light the candle
Go back to the stove and light the candle. (See How to get the Scorpion key & Shotgun in Resident Evil 7?)
Access the locked door
Burn the rope locking the door beside the monitors. You will find yourself inside the balloon room.
Unlock the door with the password
Go to the door with the password inside the balloon room and enter the password ‘loser’. This password was revealed earlier on the tape you found.
Get the valve handle
Go inside the room and grab the valve handle. (See The three Dog heads location in Resident Evil 7)
Stop the water
Go back to the room with the birthday cake and use the valve on the pipe to stop water from flowing from the ceiling.
Light the candle again
Light the candle again using the stove.
Put the candle on the cake
Go back to the birthday cake and place the lit candle.
Grab the bomb
You will find a bomb next to you on the floor. Grab the bomb quickly.
Throw the bomb through the hole
Run to the hole you created earlier and throw the bomb through it. Run away from the explosion. The wall will explode revealing a path.