Tensor networks are mathematical representations of quantum many-body systems.
Persistent homology is a method for computing topological features.
Are these two related?
It has at least two implications that I could think of in the context of renormalization groups and quantum computing & quantum complexity theory.
While the absence of evidence does not necessarily imply the evidence of absence, I would say, that currently the best answer to the stated question is:
Very little connection, besides the fact that both notions illustrate the power of abstract mathematical thinking when applied to other fields of science.
I will focus exclusively on the mathematical side and ignore the applications.
Persistent homology starts with a metric space $(X,d)$; in most applications, $X$ is a finite subset of ${\mathbb R}^n$ (for some $n$) and $d$ is the restriction of the distance on ${\mathbb R}^n$ given by a suitable norm, say, the Euclidean metric, for concreteness. As such, $(X,d)$, regarded as a topological space, might be very "boring", for instance, it has discrete topology (in most applications). The persistent homology is a mechanism for assigning some nontrivial topological invariants to the metric space $(X,d)$, which capture its metric rather than topological properties. The first step in such assignment is to introduce a family (parameterized by positive real numbers $R\in [0,\infty)$) of simplicial complexes $X_R$, called Vietoris-Rips complexes. Roughly speaking, $X_R$ captures "clustering" on the scale $R$ of points in $(X,d)$. (Formally speaking, vertices of $X_R$ are the elements of $X$. Edges of $X_R$ are unordered pairs of points $x, y\in X$ satisfying $0< d(x,y)\le R$. Each complete subgraph $K_{n+1}$ of the resulting graph then is "filled in" with an $n$-dimensional simplex.) Each complex $X_R$ is a topological space, so one can compute its homology groups $H_*(X_R)$ (with some coefficients, frequently, ${\mathbb Z}_2$; I will suppress the choice of coefficients). Then one "studies" how these homology groups behave as a function of $R$. Such study is the "persistent homology theory." What one sees here is a combination of metric geometry and algebraic topology: The input is metric, the output is algebraic (a family of homology groups or, frequently, just of their ranks, called Betti numbers).
A tensor network $T$, in contrast, has no metric structure or a natural parameter to consider. It is an oriented multi-graph (a quiver), whose edges $e$ are decorated by (specified) vector spaces $W_e$ and whose vertices $v$ are decorated with certain (yet to be specified) multi-linear maps from tensor products of the vector spaces $W_{e-}$ labelling the "incoming" edges to tensor products of the vector spaces $W_{e+}$ labelling the "outgoing" edges $e+$. Thus, the object here is a certain mix of algebra and graph theory. At this point, there is no topology. One can then associate with each $T$ its "moduli space," ${\mathcal M}(T)$. Informally, this moduli space collects all possible multi-linear maps for the vertex spaces and quotients by the automorphism group of the network.I will not attempt to give a detailed definition here; one can read, for instance
Vasily Pestun, Yiannis Vlassopoulos, Tensor network language model, arXiv:1710.10248.
Remark. For the record: The authors of the paper add one extra layer of complexity to the picture: They equip each vector space $W_e$ with a "metric," which should not be confused with a metric $d$ in Part 1 of my answer. For them a "metric" is a hermitian metric on a complex vector space $W_e$. I will ignore all this in part, since tensor networks are more general than the "isometric" tensor networks they are considering.
Now, the moduli space ${\mathcal M}(T)$ can be viewed as a topological space; sometimes it is a manifold (but not always!). However, regarding ${\mathcal M}(T)$ just a topological space is "morally wrong": The space should be treated as an algebro-geometric object (a variety or, better, a scheme, or, even better, a stack). However, to simplify things, one can treat ${\mathcal M}(T)$ just as a topological space, then compute its homology, Betti numbers, etc. This represents a mild similarity with persistent homology: Once you have a topological object, you can compute its topological invariants.
To conclude: Both persistent homology and tensor networks represent ubiquity of topology in modern mathematics. While this pleases me, as a topologist, it does not mean that any two randomly chosen examples of such ubiquity are related.