I think it is helpful to remember that there are basic differences between the commutative and non-commutative settings, which can't be eliminated just by technical devices.
At a basic level, commuting operators on a finite-dimensional vector space can be simultaneously diagonalized [added: technically, I should say upper-triangularized,
but not let me not worry about this distinction here], but this is not true of non-commuting operators. This already
suggests that one can't in any naive way define the spectrum of a non-commutative ring.
(Remember that all rings are morally rings of operators, and that the spectrum of a commutative ring has the same meaning as the [added: simultaneous] spectrum of a collection of commuting operators.)
At a higher level, suppose that $M$ and $N$ are finitely generated modules over a
commutative ring $A$ such that $M\otimes_A N = 0$, then $Tor_i^A(M,N) = 0$ for
all $i$. If $A$ is non-commutative, this is no longer true in general. This reflects the fact
that $M$ and $N$ no longer have well-defined supports on some concrete spectrum of $A$.
This is why localization is not possible (at least in any naive sense) in general in the non-commutative setting. It is the same phenomenon as the uncertainty principle in quantum mechanics, and manifests itself in the same way: objects cannot be localized at points in the non-commutative setting.
These are genuine complexities that have to be confronted in any study of non-commutative geometry. They are the same ones faced by beginning students when they first discover that in general matrices don't commute. I would say that they are real, fascinating, and difficult, and people have put, and are currently putting, a lot of effort into understanding them. But it is a far cry from just generalizing the statements in Hartshorne.
The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving".
Best Answer
René Thom's theory of morphogenesis involves singularities, unfoldings, perturbations of analytic/geometric structures, etc., which, in its turn, involves (or, rather, should involve, as the whole theory is rather sketchy) a good deal of commutative algebra.
A conference "Moduli spaces and macromolecules".
Some biological models involve systems of boolean equations, or sentences of propositional calculus, which could be interpreted as polynomials over GF(2), with subsequent application of Gröbner basis technique. A (more or less random) sample of possibly relevant papers (I avoid mentioning algebraic statistics which was mentioned many times elsewhere):
G. Boniolo, M. D'Agostino, P.P. Di Fiore, Zsyntax: A formal language for molecular biology with projected applications in text mining and biological prediction,PLoS ONE 5 (2010), N3, e9511 DOI:10.1371/journal.pone.0009511
A.S. Jarrah and R. Laubenbacher, Discrete models of biochemical networks: the toric variety of nested canalyzing functions, Algebraic Biology, Lect. Notes Comp. Sci. 4545 (2007), 15-22 DOI:10.1007/978-3-540-73433-8_2
R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, J. Theor. Biol. 229 (2004), 523-537 DOI:10.1016/j.jtbi.2004.04.037 arXiv:q-bio/0312026
I. Lynce and J.P. Marques Silva, Efficient haplotype inference with boolean satisfiability, AAAI'06, July 2006; SAT in Bioinformatics: making the case with haplotype inference, SAT'06, August 2006; http://sat.inesc-id.pt/~ines