You can do the next best think. To a  Forman-Morse function $f$ one can associate  a flow on the manifold whose stationary points are precisely  the barycenters  of the faces  of your simplicial decomposition.    The Conley index of the barycenter of a critical face has the homotopy type of a sphere of the dimension of that face. The Conley index of the barycenter of a non-critical face is homotopically trivial.

Additionally, one can construct a continuous function $\tilde{f}$ on the manifold that decreases along the trajectories of this flow and whose value at a barycenter is equal to the value of $f$ on the corresponding face.

For details  see Chapter 11 of [this paper][1].  The faces of the barycentric subdivision of your simplicial complex are invariant  sets of this flow, and on  such a face the flow is depicted in Figure 2, p.16 of the above paper.

It took me a while to realize that in Morse theory  the gradient flow associated to a Morse function is more important than the function itself.  The function plays a sort of accounting role and  the Morse condition  restricts the nature of the stationary points of the gradient flows.

**Remark**  A while ago I asked [this question][2] on MathOverflow that is related to the abundance of discrete Morse functions. They are extremely rare as opposed to the  usual smooth Morse functions that are generic.


  [1]: https://www3.nd.edu/~lnicolae/tameflow.pdf
  [2]: https://mathoverflow.net/questions/86193/combinatorial-morse-functions-and-random-permutations