Do conservative forces obey Newton's laws of motion？

Well all known forces obey newtons law . Where you may get confused is that the acceleration of each particle is not constant because their distance is changed.

Is the force between two charge particles a counterexample?

The force between two charged particles does require a generalization of Newton’s laws, in particular Newton’s third law.

Coulomb’s law respects Newton’s Third law, as long as the charges are static the forces are equal and opposite. However, once the charges begin moving magnetic forces are also involved and the forces are no longer equal and opposite.

The generalization is to realize that the general principle behind Newton’s third law is conservation of momentum. If you start with conservation of momentum then you can derive Newton’s third law for mechanical systems, but for the force between two charges particles the conservation of momentum holds while Newton’s third law does not. This is because the electromagnetic field carries momentum in addition to the charges.

Your question is far from being trivial and consequently the answer it is not.

First of all, one has to slightly decouple the concept of conservative force and the exact form of equation of motion used in connection with that force.

Strictly speaking, a conservative force is a force such that the corresponding work evaluated over every possible path depends only on the starting and final point and not on the path. Such a property allows to derive many interesting results, one of them being the possibility of coding the whole information about the vector forces ${bf F}_i$ into a single scalar quantity, the potential energy $U({bf r}_1,dots,{bf r}_N)$ such as $$ {bf F}_i=-nabla_{{bf r}_1}U({bf r}_1,dots,{bf r}_N). $$ It is true that the work-energy theorem holds only provided the equation of motion follows the second Newton's law. But it still possible, and it is currently done, to use conservative forces even with equations of motion different from $ ddot{bf r}_i= {bf F}_i/m_i$, like for example the case of Langevin's equation where, in addition to the usual force there is a random force modifying the mathematical structure of the second law.

A part the case of electromagnetic interactions, already cited in other answers, where the third law in its original form is violated, there is at least another case where the third law (again in the original form) is not obeyed in an inertial frame, while the forces are still derivable from a potential. It is the case of a 3- or many-body interaction, not amenable to a sum of pair-wise interactions.

An other case of breaking of the connection between conservative forces and second Newton's law, is the case of modifications of the second law. An interesting exercise in this direction is provided by the modification of the equation of motions proposed by Modified Newtonian dynamics (MOND) theory, although, at the best of my knowledge there is no consensus about the validity of the proposal.