It strongly depends on what you mean by "geometric interpretations". For example, normal affine varieties $X = Spec R$ where $R$ is $F$-rational have pseudo-rational singularities, and F-rational singularities deform in the sense of @NeilEpstein excellent comment. This can be done by calculating the parameter test submodules of the canonical module. Does this count for you as a deformation? As geometric?

In some ways, while these test submodules are coming from tight closure, they are also interpretable without mentioning the closure operation itself, and so its a bit unclear if what you are asking about is really tight closure or more broadly Frobenius methods.

It is important, I think, to clarify also what you mean in the sense of deformation. By this paper by Schwede and Zhang, one needs to be careful as Bertini theorems usually used in classic deformation theory for complex varieties aren't available for all classes of F-singularities -- notably F-injective singularities.

So to summarize there is a bit of tension. The usual thing one wants to do is consider a ring as a total space for a deformation and see that the singularities along a special fiber R/fR for f a regular element, extend near by fibers. This happens in two steps, first show the total space has at worst those types of singularities, and then show via Bertini that the near by fibers do too. For F-regular, its possible we can't extend to the total space in the first place and for other classes which do extend, like Cohen-Macaulay F-injective singularities, we won't have a Bertini statement available.

If you can give a more precise version of your question, maybe someone can say more.

It strongly depends on what you mean by "geometric interpretations". For example, normal affine varieties $X = Spec R$ where $R$ is $F$-rational have pseudo-rational singularities, and F-rational singularities deform in the sense of @NeilEpstein excellent comment. This can be done by calculating the parameter test submodule of the canonical module. Does this count for you as a deformation? As geometric?

In some ways, while these test submodules are coming from tight closure, they are also interpretable without mentioning the closure operation itself, and so its a bit unclear if what you are asking about is really tight closure or more broadly Frobenius methods.

It is important, I think, to clarify also what you mean in the sense of deformation. By this paper by Schwede and Zhang, one needs to be careful as Bertini theorems usually used in classic deformation theory for complex varieties aren't available for all classes of F-singularities -- notably F-injective singularities.

So to summarize there is a bit of tension. The usual thing one wants to do is consider a ring as a total space for a deformation and see that the singularities along a special fiber R/fR for f a regular element, extend near by fibers. This happens in two steps, first show the total space has at worst those types of singularities, and then show via Bertini that the near by fibers do too. For F-regular, its possible we can't extend to the total space in the first place and for other classes which do extend, like Cohen-Macaulay F-injective singularities, we won't have a Bertini statement available.

If you can give a more precise version of your question, maybe someone can say more.