The chapter below of the Assassin's Creed III Remastered guide contains a walkthrough for sewers in New York.
To open the door, move the left analog stick left, the right twice to the right and afterwards the left twice more times to the left.
To open the gate, move the left stick twice to the left, the right stick to the right and afterwards the left three times to the left.
Nearby the Brewery exit you should see a gunpowder barrel.
Place it beside the wooden barricade and blow it up to unlock another tunnel.
Get rid of it just like obstacle number 3.
Place the lens in the following order: eye above, weight on the right, the below and the cross on the left.
Go round the grate from the left by jumping between rocks.
Turn the spinning wheel to lower the lift and afterwards jump through the grate to open it from the other side.
Jump through the second door as well and turn the spinning wheel. That way you will open the gate. The third passage can be unlocked by going round it from the left. from the wooden barrel, run onto the wall and jump further straight and to the right.
To open the door, place the lens in the following order: eye above, crown on the right, letter below and the eagle on the left.
To get pass the grate, you need to jump between two walls and avoid the falling water. Start off by climbing onto the wall on the left, hold down the free run button and press jump. Proceed doing so until you teach the top.
To open the door, move the left analog stick to the left, the right to the right and afterwards the left to the right. That way you will turn the valves in the proper way.
To get past this obstacle, jump right from the left wall and blow up the barrel on the other side of the room. That way you will unlock a passage behind the barricade and will be able to open the door. Get past the next obstacle in the same way.
To open the door, place the lens in the following order: sun above, letter on the right, weight below and circle on the left.