Magnetic levitation has been used to suspend frogs in midair. I was wondering: Does the animal still feel gravitational pull? I mean: Does the frog feel like an astronaut on the ISS, or does he feel like a trapeze artist suspended by a harness?
You would feel weightless if every part of your body of mass $m$ would be subject to an upward force equal to $m$ times the local gravitational acceleration $g$. Such an exact part-by-part cancellation is not going to happen via diamagnetic levitation as utilized on the frog in your example. Not only does this levitation couple according to magnetic susceptibility, and not to mass, but more importantly, such levitation relies on inhomogeneous magnetic fields. This means that one or more (central) parts of the frog get pulled up more strongly than other parts.
My guess is that the frog feels less weight but strangely suspended by its stomach.
The feeling of weight is caused by contact forces acting on only parts of your body: forces like the pavement pushing on your feet, air rushing past certain parts of your body as you fall at terminal velocity, the floor of an elevator pushing with reduced force on your feet as the elevator starts its downward trip. Basically you feel one part of your body pressing or pulling on other parts to keep them all together.
Gravity is a distributed force. It acts on each particle of mass identically (with rare exceptions: See Larry Niven's, Neutron Star) Your body does not need any internal forces to keep all the parts together, and you feel "weightless".
With regard to floating submerged, you are suspended by the difference in hydrostatic pressure at the top and bottom of each element. Buoyancy is volume-based, not mass-based. To the extent that there are variations in density in your fluid filled body, you will feel weight floating in water. Consider, for example, the crew-members in a (non-flooded) submarine.
With regard to magnetic levitation, Johannes' answer above is correct. To the extent that various masses in the frog's body experience magnetic forces not in proportion to their mass, the frog will experience "weight".
The trapeze harness feeling comes because even though you are suspended, the harness is exerting a normal force on you. an easy way to think of this is that you still have gravity pulling all of your organs down. The frog, on the other hand, experiences none of that. The electromagnetic force is acting on all parts inside and outside of its body; it completely cancels gravity. Thus, it would feel weightless.
There are different components to feeling weightless.
You can partially simulate weightlessness with neutral bouyancy in water. This makes gross physical activity feel like it would when weightless (well: weightless and immersed in a dense, viscous medium).
Furthermore, divers have great difficulty sensing "down" even when bouyant, so in that sense also they "feel" weightless (at least, they fail to feel their weight). Any form of support that's spread as evenly over the body as the force from an external fluid would feel about as weightless as that.
However, it's possible for a diver to have a sense of "down" from e.g. the inner ear or the organs. Humans shouldn't count on that to get to the surface in the dark, I don't know whether frogs can. An accelerometer in an air bubble in a box would of course work even if the overall gadget is neutrally bouyant. I believe that training underwater is not a good predictor of how dizzy/nauseous you'll feel in freefall, so clearly the feeling is different even if not obviously so. In some sense a highboard diver has more claim to feeling weightless than a deep sea diver -- aside from air resistance anyone falling does at least have uniform forces throughout their body.
Now, diamagnetic levitation is not a force from an external fluid, it's internal. With sufficiently even diamagnetic levitation you'll feel at least as weightless as you do when neutrally bouyant in water and possibly more so. So if you're happy to call that "feeling weightless", the frog would feel weightless in a flat magnetic field.
But aside from the shape of the magnetic field, the levitation will not be perfectly even, because your body is not universally diamagnetic: some parts are wetter than others, or perhaps more formally magnetic susceptibility need not be uniform throughout the materials of the body. So the question is whether the frog can detect the unevenness of the levitation. If not then it feels weightless. Probably the answer is like water for humans -- it's distinguishable from freefall, but the direction of the weight is not immediately obvious.
Then again, if a particular frog is vomiting copiously in freefall but not in magnetic levitation it will tell you "sure, it feels very different". Vomiting feels different from not vomiting, regardless of how subtle was the effect on its ears that caused it to vomit!
What you feel in a harness is a force equal to your weight exerted on a few square inches that are not any of the usual few square inches "designed" for that purpose. So magnetic levitation would only feel like a harness if the levitation is primarily of a few small parts of the body, with much less or no force on the rest. I do not believe that this is generally the case, and those levitating frogs don't have their limbs dangling downward, but it's possible that it is.
So, you've got your theory and then you've got your messy practical details of the construction of strong magnets and of frogs. I suspect it really comes down to what the frog is trying to achieve, and what fine mechanisms frogs have that can detect "down".
As I mentioned in my comment, it seems you will feel weightless when the net force exerted on you is equal to the force of gravity. (So basically, gravity is the only force in play)
This is consistent with the fact that astronauts in the ISS are weightless even though they do experience a net force (they are being kept in a orbit around earth, after all). It also plays very nicely with people in free fall feeling weightless (until air resistance kicks in anyway), and hypothetical astronauts in deep space feeling weightless (no gravity, no net force).
This makes sense when you consider the definition of weight. Weight is the force you are exerting on the earth (or whatever connects you to the earth), not the other way around.
In the case of the frog suspended in a magnetic field (as well as hanging in a trapeze) there is no net force, but there is gravity, therefore the frog will experience his normal weight. In this case, the force on the earth is exerted via the magnetic field.