The Casimir effect causes a limited number of wave functions, causing fewer particles between plates than outside and this could be considered as a negative mass. Then, if we have a technology good enough so that the total energy of the system is negative, will the system have acceleration towards the opposite direction of a force applied to this system?
It's a good question, and the simple answer is that we don't know.
I would repeat Anna's comment that virtual particles are a calculational device and do not really exist, so you need to be careful when talking about there being fewer virtual particles between the plates. However it is definitely true that the vacuum energy density is lower between the plates than it is outside them. That is, after all, what causes the Casimir force.
The problem is that we don't know how vacuum energy gravitates, or indeed if it does. If we do a naive calculation of how much gravitational force we expect from the vacuum energy we get a ridiculous number, so clearly the naive calculation doesn't work. We have to conclude we don't understand how the vacuum energy behaves, and therefore we don't understand how the reduced vacuum energy between the plates behaves.
It is possible the region between the plates does behave as a negative mass, though the end result would simply be that the total mass of the two plates plus the space between them was slightly less than the mass of the two plates alone. And we see this sort of thing routinely as the mass of any bound system is slightly less than the mass of its parts. This is what causes the mass deficit in nuclei.