Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?
I will differ somewhat with the answers by Carlo and Zurab. Just because the identity now known as the SchwingerDyson equation was first discovered in the context of Green functions, does not mean that it is most naturally expressed in terms of Green functions (aka QFT correlation functions).
Let me refer to the excellent treatment of the SchwingerDyson equation in Chater 15 of Quantization of Gauge Systems by Henneaux & Teitelboim (PUP, 1994). Consider functionals $A[\phi]$ on the space of (offshell) field configurations $\phi$. Let us suppose that all of these functionals are sufficiently regular so that we can use the variation formula $A[\phi + t\zeta] = A[\phi] + t \int \alpha_i[\phi](x) \zeta^i(x)\, dx + O(t^2)$, for some $\alpha_i[\phi](x)$ which we also denote by $\frac{\delta A[\phi]}{\delta \phi^i(x)}$. One of these functionals is the action functional $S[\phi]$ and its variation $E_i[\phi](x) = \frac{\delta S[\phi]}{\delta \phi^i(x)}$ gives the classical equations of motion of the field theory.
For any given offshell functional, there is a map $A[\phi] \mapsto T(A[\phi])$ which stands for the time ordered quantization of $A[\phi]$, so that $T(A[\phi])$ is an element of the quantum algebra of (onshell) observables (or operators). The map $T$ is linear, but obviously not an algebraic homomorphism because it maps a classical (commutative) algebra into a (quantum) noncommutative one. Also, since it maps (classical) offshell functionals to (quantum) onshell operators, it must have a kernel that is somehow generated by the equations of motion.
The SchwingerDyson equation precisely specifies the kernel of $T$ within the space of classical offshell functionals: $$ T\left(A[\phi] \int \lambda^i(x) \frac{\delta S[\phi]}{\delta \phi^i(x)}\, dx + \frac{\hbar}{i} \int \frac{\delta A[\phi]}{\delta \phi^i(x)} \lambda^i(x)\, dx \right) = 0 , $$ for any $A[\phi]$ and $\lambda^i(x)$, where $\lambda^i$ needs to be $\phi$independent. Note that the treatment in Henneaux & Teitelboim uses equations like $\langle T() \rangle = 0$ instead of $T() = 0$. However, they use $\langle \hat{O} \rangle$ to stand for an arbitrary matrix element of the operator $\hat{O}$ (evaluated using a path integral), so the two kinds of equalities are equivalent. As can be seen from the above formula, the kernel of $T$ is an $\hbar$deformation of the subspace of offshell functionals generated by $\int \lambda^i(x) E_i[\phi](x)\, dx$ for arbitrary $\lambda^i(x)$, which all vanish when restricted onshell, to the solutions of $E_i[\phi](x) = 0$ among all possible field configurations. To get the full classical kernel of the restriction onshell, we would need to allow $\lambda^i(x) = \lambda^i[\phi](x)$ to be $\phi$dependent. But doing that naively breaks the above formula. So, when possible, arbitrary $\lambda^i[\phi](x)$ should be approximated by linear combinations of appropriate choices of $A[\phi]$ and $\lambda^i(x)$.
One final remark about a more geometric way of rewriting the above version of the SchwingerDyson equation. Let us interpret $\lambda^i(x)$ as a vector field on the field configuration space and denote the corresponding Lie derivative by $\mathcal{L}_\lambda$. The SchwingerDyson equation then reads $$ T\left( \frac{\hbar}{i} e^{iS[\phi]/\hbar} \mathcal{L}_\lambda ( A[\phi] e^{iS[\phi]/\hbar} ) \right) = 0. $$
A quick and clear and rigorous derivation of the SchwingerDyson equation from BVtheory in causal perturbation theory is offered in remark 7.7 of Rejzner 16, it's spelled out at nLab:BVoperator  SchwingerDyson equation.
The upshot is the the SDequation is nothing but the incarnation of the free field theory quantum BVdifferential, schematically of the form $\{S,\}_{BV} + i \hbar \Delta_{BV}$; hence the quantum correction in the SDequation originates in the BVoperator $\Delta_{BV}$.
In the traditional pathintegral heuristics this is highlighted already in section 15.5.3 of HenneauxTeitelboim 92 (which Igor Khavkine is pointing to in his reply), where therefore it is suggested to call the free quantum BVdifferential the "SchwingerDyson operator".
I'm not sure why you would want to avoid Green's functions, which provide the most natural formulation of the SchwingerDyson equations, but indeed, algebraic approaches do exist. In the context of random Riemann surfaces the SchwingerDyson equations appear as a set of algebraic constraints on the partition function, and they can be given a geometric interpretation. See for example Section III of The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders.
In the formulation of QFT using formal functional integrals, as mentioned by Igor in his answer, the SchwingerDyson equation becomes an infinitedimensional differential equation for the partition function. As such, since the partition function is the generating functional for the Green functions, all you are doing is simply grouping the SchwingerDyson equations for all Green functions together in a single formula. I also share Igor's viewpoint that the SchwingerDyson equations written in terms of the partition function is more natural, because it makes the structure of the equation much more evident.
That being said, there are two nice references that have not been mentioned by the answers given so far. One is the book Path Integral Methods in Quantum Field Theory by R. J. Rivers (Cambridge University Press, 1987), where the SchwingerDyson equations are a central concept and are formulated in many different ways. Another, which I strongly recommend, is the paper by M. Dütsch and K. Fredenhagen The Master Ward Identity and Generalized SchwingerDyson Equation in Classical Field Theory, Commun. Math. Phys. 243 (2003) 275314, arXiv:hepth/0211242. In this second reference, it is shown that the SchwingerDyson equation is actually a particular case of the quantum correction (due to perturbative renormalization) to all identities that follow from the classical equations of motion, called the Master Ward Identity (MWI for short). The quantum BVequation mentioned in Urs's answer is also a particular case of the MWI, which has a nice algebraic interpretation  to wit, one can see the classical limit of the MWI as the simple fact that the (lefthand sides of) all identities that follow from the classical equations of motion are elements of the ideal of offshell functionals of field configurations (satisfying certain conditions which we do not need to recall here) which is generated by the EulerLagrange operator. At the quantum level, not all identities survive due to renormalization  the ensuing violations are known as anomalies. This means that the "classical" ideal of formal power series in $\hbar$ of functionals generated by the EulerLagrange operator is no longer an ideal with respect to the quantum product. However, the most general form for the anomalies is dictated precisely by the MWI. In the case of local gauge (or BRS) symmetries, this leads through the quantum BV equation to the WessZumino consistency conditions and so on.
Just a final word of warning: it must be said that all of the above can be made rigorous only at the formal perturbative level. In constructive QFT, the SchwingerDyson equations are usually not used as a tool to construct a model but are rather derived rigorously as a consequence after the model has been constructed, since only then can they be made sense of. Analysis of QFT models using the SchwingerDyson equations is usually deemed as "nonperturbative" for the equations make no explicit mention of perturbation theory, but making sense of this equations outside perturbation theory is a messy business.
In addition to Carlo Beenakker's answer, in the context of the theory of random matrices the SchwingerDyson equations are discussed (without any mention of Green's functions) in the book http://link.springer.com/book/10.1007%2F9783540698975 (Large Random Matrices: Lectures on Macroscopic Asymptotics, by Alice Guionnet). See also http://arxiv.org/abs/1307.1806 (SchwingerDyson equations: classical and quantum, by James A. Mingo and Roland Speicher).
However, I agree to Carlo Beenakker that the most natural language for SchwingerDyson equations in quantum field theory is the language of Green's functions. Even when the SchwingerDyson equations are discussed in the very abstract mathematical context, it is useful to use the analogy with Green's functions to get an insight of what is going on. See, for example, http://arxiv.org/abs/hepth/0404090 (The residues of quantum field theory  numbers we should know, by Dirk Kreimer) and http://arxiv.org/abs/hepth/0407016 (What is the trouble with DysonSchwinger equations? by the same author).

1$\begingroup$ Thanks for the responses, especially Igor's since H&T is by my side right now and is in the context that provoked the question. quoting `` it is useful to use the analogy with Green's functions to get an insight of what is going on'' applies only if one has some feeling for Green's functions $\endgroup$ – Jim Stasheff Oct 7 '14 at 13:46