What is the difference between adiabatic and nonadiabatic tunneling for a molecule in a laser-field? Does nonadiabatic tunneling necessarily relate to the Conical intersection or curve-crossing?
This is very far from a complete answer, especially w.r.t the part of your question regarding "laser fields". After doing some reading, nonadiabatic tunneling is conceptually the same as adiabatic tunneling except that one allows for coupling to another adiabatic state.
Adiabatic tunneling is the phenomenon where a particle described by a wavefunction is propagating on a potential and begins with less energy than the height of the barrier it approaches. One finds, however, that there is amplitude of the wavefunction on the other side of the barrier despite the particle having insufficient energy to cross the barrier.
Now, in physics and chemistry, nonadiabatic processes typically describe those processes where it is a bad approximation to consider only a single potential energy surface. That is, both surfaces may be adiabatic states, but these states are coupled in some manner. In general, there will always be some small coupling between states that are not too far apart in energy.
Nonadiabatic tunneling just describes the situation that one does not neglect this coupling between states. To a first approximation, nothing will change and one will find the same amplitudes as a function of position and time. If you consider the presence of the higher-energy adiabatic state as a perturbation of the Hamiltonian of the original particle, then the zeroth order picture will just be adiabatic tunneling. The next term will be something like the case where the particle is either reflected or transmitted over the barrier, but it does so by first "visiting" the higher-energy adiabatic state.
Now, with that description in mind, I think it is reasonable to say that nonadiabatic tunneling does not necessarily have to be involved with with conical intersections or curve crossing. Nonadiabatic tunneling is just regular tunneling which allows other states to couple with the original state. Now, of course, conical intersections and curve crossing could very well come into play in nonadiabatic tunneling because this would just be another form of coupling one could introduce.
As an aside, the description I give where the higher-energy state is a perturbation on the original system is probably not a good one in general as one could imagine very strong coupling between the potentials (as would be the case with a conical intersection).