The passengers only feel the normal force, which is always directed 'up,' through the roof of the plane. So there are 2 forces acting on the plane that we are concerned with. The force of gravity acting down, and the lift force acting up through the roof of the plane. For the passengers inside, the lift force is replaced by the normal force, which has exactly the same direction and magnitude.
Note: The centripetal force shown in the image is not real. It is just a component of the lift force.
As you can see, the lift force is greater in magnitude than the weight. Thus, the passengers will experience an increased 'load'; that is, they will feel heavier. That is because the normal force (or lift force), which is normally equal to the force of gravity, $mg$, is now equal to $\alpha mg$, where $\alpha$ is the load factor, usually measured in "G's". If the plane is tilted at 60 degrees to the vertical, the plane feels twice as heavy. In other words, the plane is "pulling 2G," i.e. $\alpha=2$
Your question specifically asks whether the passengers experience the feeling of inclination. This feeling comes from the fluids in our ears - our internal accelerometers. That is where we get the perception of being 'right-side up' or 'rolled sideways'. Unfortunately, our internal senses go out of sync if we do not have a reference to the horizon. That's why it is critical for pilots to put complete faith in their instruments (altitude indicator, altimeter, airspeed, etc.) when flying in instrument meteorological conditions, because their senses will lie to them. I highly recommend if you are interested in this, to refer to the FAA Instrument Pilot training manual, which covers this important topic: The relevant material starts on page 3-4
In response to your comment:
Using solely the rudder is a very inefficient way to turn an airplane. The reason why the wright brothers were so successful with their Flyer was that they mastered control; i.e. before they slapped an engine onto their glider, they figured out how to maneuver it. They invented the 'wing-warping', adapted form the flight of birds, which gracefully rolls the plane in one direction by changing the lift on both left and right sides of the wing (this also creates an adverse-yaw in the opposite direction!). I'm afraid I can't provide you with the reference, where the wright flyer is compared with a european model that had no wing-warping capability or ailerons, and it turned exclusively with the use of rudder. Suffice it to say, the wright flyer won out.
Note: There aren't (none!) planes that turn only with rudder! The rudder is used to make a coordinated turn (among other things), NOT as the primary tool to turn the plane. That is what ailerons are for (and elevator to keep you level)!
One last thing I want to point out is that adding more rudder to help you turn can very will KILL you. If you are on base and turning to final leg of the approach to the airport, you will be flying slow. If you feel you are overshooting the turn, you will be tempted to turn tighter. However, as you turn tighter, you slow the airplane down faster, and begin approaching the stalling speed (you really shouldn't be, you should be some safe margin faster than that). Let's suppose that you can't bank any more, so you decide to add a little more left rudder. This is the perfect scenario for a spin-stall. At such low altitudes (<1000 ft AGL), you wouldn't be able to recover. The moral of the story is: do NOT treat a plane like a ship in air. The plane turns by banking, the steeper the bank, the tighter the turn. The tighter the turn, the faster you bleed off energy (speed). If you lose too much energy, you will fall out of the sky. Don't make that happen when you are too low to the ground (you won't have any potential energy to convert back to kinetic!)
I found the reference I was looking for: Anderson's Introduction to Flight, 3rd Ed. It is the first page of chapter 7, Principles of Stability and Control. Here's an excerpt: "In a struggling circular turn, Farman deflects the rudder and mushes the biplane around the marker, the wings remaining essentially level to the ground. Continuing in its rather wide and tenuous circular arc, the airplane heads back." Later on in the text: "Wilbur takes off. Using the Wrights' patented concept of twisting the wingtips, Wilbur is able to bank and turn at will. He makes two graceful circles..."