Can we have motion in systems where inertia is neglected? According to Newton's law:
$$ \sum F=ma$$
So, if we have some acceleration, it's because we have a certain amount of motion in our systems.
This makes me confused if inertia was neglected. Are there cases where we can have motion when inertia is neglected? Maybe when there is a friction so we have velocity, but this friction will be already balanced with other forces, so it shouldn't move. Or viscous systems?
Adding to the question after reading the comments:
Can example about that: cells are viscoelastic media, there, the inertia can be neglected with respect to viscous forces, but still the cell can move. According to biological processes happens in the cells, we must worry about the stresses at the surfaces coming from for example the actin-myosin contractility. So we can say that motion comes from deformations?
Also, if we have a viscous material or rod so that we can neglect inertia, and we pull this material, we can say that $F_{ext}=\text{rate of deformation} \times viscosity$, and the rate of deformation depend on the velocity and the shape of the rod will involve with time?
 A: I'm not completely sure I fully understand your question, but I'll try to give you an answer...
It might help to think of this problem in a pseudo-relativistic context, wherein the physical motion of objects in inertial reference frames (i.e. relative to a stationary observer or one moving at constant velocity $v_0$) all behave similarly, cf. Newton's 1st law of motion. That is, one obtains exactly the same equations of motion whether one conducts an experiment at rest ($v_0=0$) or moving with constant velocity ($v_0\neq0$), e.g. in a lab or on a train, say (albeit different solutions).
It is always possible to change one's point of view/switch reference frame by utilizing different coordinate transformations; such as the classical Galilean transformation $x\rightarrow x'=x+v_0\cdot t$ or with the relativistic Lorentz transformation $x\rightarrow x'=\gamma\cdot(x+v_0\cdot t)$. This effectively means, that the relative kinematics can generally be absorbed into some (re)parameterization of the coordinates, yielding the exact same dynamics! For instance, in the aforementioned transformations/mappings from rest frames to co-moving reference frames, the coordinates are parameterized by the relative velocity $v_0$. 
The complications usually arise when inertial reference frames are transformed into non-inertial reference frames (i.e. accelerating relative to an inertial observer). But keep in mind, that there are some important distinctions between the relative measurements and the actual physics occuring (the major difference being of ontological nature). For example, the acceleration caused by frictional forces exerted on an object placed on a train table is completely independent of the inertial motion of the train ($\Delta v_0=0$), and only dependent on the train's non-inertial motion ($\Delta v_0\neq0$).
So, I think the answer you are looking for is "yes, inertia can be neglected in the equations of motion for dynamical systems".
A: It is not clear what is meant by inertia is neglected:

*

*Small/huge inertia  It could be that the inertia is so small, that particle experience huge acceleration. We do often use this kind of an approximation - e.g., when considering elastic collisions against walls, and claiming that the particle with momentum $\mathbf{p}$ is reflected with momentum $\mathbf{p}$. Newton's equation tells us that
$$
\frac{d\mathbf{p}}{dt}=m\frac{d\mathbf{v}}{dt}=m\mathbf{a}=\mathbf{F},
$$
i.e., we are dealing here with an infinite force that changes the particle momentum and direction of motion instantaneously. In fact, this is an approximation for a particle of mass $m$ scattered against a much heavier object (i.e., the wall) of mass $M$, valid when $m/M\ll 1$.

*Cancellation of mass In some problems the force is itself proportional to a mass, so that the mass cancels out from the Newton's second law equation. E.g., a body sliding along an inclined plane in presence of friction obeys
$$
ma=mg\sin\alpha - \mu mg\cos\alpha\Rightarrow a = gsin\alpha - \mu g \cos\alpha
$$
Similar equations arise when dealing with viscous friction.

A: We sometimes assume a body has negligible mass, such as a rope in a pulley system where the pulley and weights on the ends of the rope have much greater mass than the light rope.  This simplifies the evaluation of the motion. See a basics physics textbook for examples.
