We consider goal-oriented adaptive space-time finite-element discretizations of the parabolic heat equation on completely unstructured simplicial space-time meshes. In some applications, the main interest lies in an accurate computation of some possibly nonlinear functionals at the solution, so called goal functionals. This motivates the use of adaptive mesh refinements driven by the dual-weighted residual (DWR) method. The DWR method requires the numerical solution of a linear adjoint problem which provides information on where the error in the solution has the most influence on the goal functional. This can be done by means of the same full space-time finite element discretization as used for the primal linear problem. The numerical experiment presented demonstrates that this goal-oriented full space-time finite element solver efficiently provides accurate numerical results for a model problem with moving domains and a linear goal functional.