Is the deformation of spacetime, elastic deformation or plastic deformation?

I would argue that it is an elastic deformation because when a mass moves from place to place the "deformation" to spacetime does not persist.

@peterh made an interesting point when bring up the energy loss due to gravitational waves in a many-body system (like the Sun+Earth). But for a single body in space moving at a fixed velocity, say a rogue star without a solar system, it will not radiate gravitational waves (see below). But this star will still locally deform spacetime and that deformation will not persist after the star passes.

According to Wikipedia:Gravitational wave sources,

In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or rotationally symmetric (like a spinning disk or sphere). A simple example of this principle is a spinning dumbbell.

The answer of your question is quite deep.

Lets begin with a very basic of elastic deformation in continuum mechanics, where people starts with defining the "deformation" of field as $u^{\alpha}(x)\equiv \bar{x}^{\alpha}-x^{\alpha}$. Which is in fact indicates how each point in a solid moves under a deformation.

The deformation contributes to the thermodynamic functionals like free energy, entropy etc. To the lowest order, the thermodynamical functionals will be quadratic in the scalars constructed from the derivatives of the deformation field. The extremisation of the relevant functional (entropy, free energy ....) allows one to determine the equations which govern the elastic deformations.

The analogue of elastic deformations in the case of spacetime manifold will be the transform as $x^{a}(x)\rightarrow \bar{x}^{a} =x^{\alpha}+v^{a}(x)$. If we accept the general coordinate transformation as elastic deformation of spacetime then one can derive the consistency of elastic condition deformation provided that Einstein field equations are satisfied! In this respect we can say that spacetime behaves like a solid, which can undergo elastic deformation.

For a complete derivation please check this paper Gravity as elasticity of spacetime

Extending @LasersMatter : I would also vote for a nearly perfectly elasticity.

A little bit of the energy is lost due to gravitational waves. Thus, the process can't be reversible perfectly. We can say, the gravitational waves radiated away have an entropy.

It is very little in nearly all settings: for example, the spacetime deformation of the Earth-Sun system loses only around 200W due to them. In the black hole merge which was found recently by the LIGO, around 3 Solar mass was lost from the 60 Solar masses of the black holes.