What is a resistor?
A resistor exhibits useful resistance, which is a way of saying it obeys Ohm's law.
"Useful" meaning it obeys Ohm's law over a useful range of frequencies and voltage or current.
No resistor is perfectly ohmic. All real resistors exhibit dominant capacitance, inductance or more complicated impedance responses at high frequencies (typically MHz to GHz). (The frequency response is linear, so it's not strictly non-ohmic, because V(f) = I(f) * Z(f) and V and I are linear in the usual sense, but Z(f) is not a constant resistance R, rather a complex function of f itself.) Likewise, they aren't ohmic in the static sense: excessive voltage or current will cause overheating, changing the value, or burning it out, or causing arcing. Also, on a very precise scale (parts per million), resistors are nonlinear for a variety of poorly understood reasons.
So with our terms defined, let us consider the diode.
A diode is strongly nonlinear, and also has complicated time-dependent properties, depending on type.
As w2aew has illustrated, PIN diodes exhibit useful resistance, at high frequencies, that is variable with bias.
Statically, a diode would not usually be said to have useful resistance: examples like the diode VGA (variable gain amplifier) or compressor that xfs mentioned are very limited in scope. The change in signal current must be a small fraction of the bias current, otherwise the signal is distorted terribly. This is using calculus, in spite the properties of the diode, not for any benefits of it. Simply put: for suitable considerations (differentiability), for an ever-smaller signal, the V(I) curve looks ever flatter (linear). But this does not change the fact that, on the whole, the diode has an I ~= exp(V) curve, which is one of the most nonlinear functions regularly encountered in math and science.
At high currents, it is true that a diode exhibits resistance: there is parasitic resistance, from the leads, and the bulk semiconductor leading up to the junction itself, which behave ohmically. At high currents, where the voltage drop across this resistance dominates over the junction voltage drop, the diode can be said to be resistive. This is important under surge conditions, where the resistance acts to limit peak current (and also acts to dissipate power, causing the diode to fail). It is not part of normal operation, because the current level is normally much higher than the diode's ratings.
Some exceptions include old fashioned germanium point-contact diodes (where the point contact has so little area that the resistance is large), and SiC schottky diodes (where the bulk semiconductor simply cannot be made as conductive as other materials, and therefore the resistance dominates at currents within ratings).
Because of these special-case conditions, it might not be genuine to say that a diode never exhibits resistance, but a diode definitely cannot be said to be a generic, unequivocal resistor.
Tim