Does the stress-energy tensor contain the equations of motion? Derivatives $\nabla_i T^{ik}=0$ of a stress-energy tensor of physical system express conservation laws. Whether contains a stress-energy tensor also the information on the equations of motion of system?
 A: 1) Let us assume that the system is classical and has a Lagrangian formulation
$$ S[\phi]~=~ \int \mathbb{L}, \qquad\qquad \mathbb{L} ~=~{\cal L} ~dx^0 \wedge \ldots\wedge  dx^{d-1}, $$
in terms of a Lagrangian density ${\cal L}={\cal L}(\phi,\partial\phi)$ that doesn't depend explicitly on the $d$-dimensional spacetime coordinates $x^{\mu}$.
2) Noether's Theorem for the global translation symmetry $x^{\mu}\to x^{\mu}+\delta x^{\mu} $ yields the following off-shell relation 
$$ d_{\mu} T^{\mu}{}_{\nu} ~=~ - \phi^{\alpha}_{,\nu} ~\frac{\delta S}{\delta\phi^{\alpha} }, \qquad\qquad 
\phi^{\alpha}_{,\nu}~:=~\frac{\partial\phi^{\alpha}}{\partial x^{\nu}} . $$
Thus one may deduce the conservation law 
$$d_{\mu} T^{\mu}{}_{\nu}~\approx~ 0$$ 
from the equations of motion (eom) 
$$\frac{\delta S}{\delta\phi^{\alpha} }  ~\approx~ 0. $$
However, there are many examples where one can not deduce the other way, in particular, if there are more eoms (labeled by $\alpha$) than conservation laws (labeled by $\nu$). [Here the $\approx$ sign means equality modulo eom.] 
3) OP raises an interesting question whether one can deduce the equations of motion from knowledge of the stress-energy tensor $T^{\mu}{}_{\nu}$ itself? This is almost the case, but there are certain terms that the stress-energy tensor $T^{\mu}{}_{\nu}$ cannot see, see Section 4, 5 and 6 below. The question is easiest to solve if one has direct access$^1$ to the canonical stress-energy tensor 
$$T^{\mu}_{({\rm can})\nu}
~:=~\phi^{\alpha}_{,\nu} ~\Pi^{\mu}_{\alpha}-\delta^{\mu}_{\nu}~{\cal L},\qquad\qquad 
\Pi^{\mu}_{\alpha}~:=~\frac{\partial{\cal L}}{\partial \phi^{\alpha}_{,\mu} }. $$
It becomes harder if there are also improvement terms allowed
$$ T^{\mu}{}_{\nu}~=~T^{\mu}_{({\rm can})\nu}+d_{\lambda}f^{\lambda\mu}{}_{\nu},\qquad\qquad f^{\lambda\mu}{}_{\nu} ~=~ - f^{\mu\lambda}{}_{\nu}, $$
and/or if one only knows $T^{\mu}{}_{\nu}$ modulo terms proportional to the eoms (with the additional complication that one doesn't know the eoms to begin with).
4) As a topological-sigma-model-like counterexample, consider a $d$-form 
$$ \vartheta ~=~ \frac{1}{d!}\vartheta_{\alpha_1\ldots\alpha_d } ~d\phi^{\alpha_1} \wedge \ldots \wedge  d\phi^{\alpha_d}, \qquad\qquad \vartheta_{\alpha_1\ldots\alpha_d }~=~\vartheta_{\alpha_1\ldots\alpha_d}(\phi), $$
in the $\phi$ target space. Let $\omega:=d\vartheta$ be a corresponding $d+1$ form. Now define the Lagrangian $d$-form
$$  \mathbb{L}~:=~\phi^*\vartheta $$ 
by the pull-back to the $x$-world volume. The corresponding canonical stress-energy tensor vanishes identically $T^{\mu}{}_{\nu} ~\equiv~0$, while the eoms become
$$\phi^*i_{\alpha}\omega ~\approx~ 0,$$ 
where $i_{\alpha}$ denotes the contraction wrt. the vector field $\frac{\partial}{\partial\phi^{\alpha}}$.
The main point is that the vanishing stress-energy tensor $T^{\mu}{}_{\nu} ~\equiv~0$ carries no information about the $d$-form  $ \vartheta$. The equations of motion are only trivial, i.e., $0=0$, if $\omega=0$, i.e., if $\vartheta$ is closed. 
Still worse, for a generic model, the Lagrangian $d$-form $\mathbb{L}$ could contain such a $\phi^*\vartheta$ term, which the stress-energy tensor $T^{\mu}{}_{\nu}$ cannot see.
5) Consider for simplicity the special case of point mechanics where $d=1$. Then the stress-energy tensor is just the energy function $h:=T^0{}_0$. Similarly, we change notation $x\to t$ and $\phi^{\alpha}\to z^I$. The one-form $\vartheta$ is a presymplectic potential 
$$\vartheta ~=~\vartheta_I~ dz^I, \qquad\qquad \vartheta_I~=~\vartheta_I(z), $$ 
and the two-form $\omega:=d\vartheta$ is a presymplectic two-form. (If the two-form $\omega$ is non-degenerated, it becomes a symplectic two-form.) The Lagrangian is 
$$L~=~\vartheta_I~\dot{z}^I,$$
which can be viewed as a Hamiltonian system with zero Hamiltonian. The corresponding energy function vanishes identically $h~\equiv~0$, while the equations of motions becomes
$$\omega_{IJ}~\dot{z}^J ~\approx~ 0.$$ 
6) To be explicit, consider $n$ non-relativistic charged particles in a magnetic vector potential ${\bf A}$. Working in units where $c=1$, the Lagrangian reads
$$L~=~\sum_{i=1}^n \frac{m_i}{2} \dot{\bf r}^2_i 
+ \sum_{i=1}^n q_i\dot{\bf r}_i\cdot {\bf A}({\bf r}_i) 
- V({\bf r}_1, \ldots, {\bf r}_n),$$
where we have added a potential $V=V({\bf r}_1, \ldots, {\bf r}_n)$ to be more general. The corresponding energy function 
$$h ~=~\sum_{i=1}^n \frac{m_i}{2} \dot{\bf r}^2_i 
+ V({\bf r}_1, \ldots, {\bf r}_n)$$
does not know about the magnetic potential ${\bf A}$. However the eoms contain the Lorentz force. Here the magnetic potential ${\bf A}$ plays the role of the presymplectic potential from Section 5.
--
$^1$ It seems admittedly rather artificial that one would know in advance whether a given stress-energy tensor $T^{\mu}{}_{\nu}$ is canonical or not, because if one doesn't know the eoms, one wouldn't know the Lagrangian density ${\cal L}$ as well. 
A: Simple answer, your equation has only four components, but there is one equation of motion for every field variable in the system which is more than four, so the answer is no.
A: The situation is similar to conservation of energy, momentum etc. in classical mechanics. Consider for instance motion of the mass $m$ in the potential $V(r)$. In general case the energy is conserved, and the momentum is not. We know the expression for energy:
$E=\frac{mv^2}{2}+V(r) = const$
The equation of motion is
$\frac{d\vec{v}}{dt}=-\nabla V(r)$
and it cannot be derived from expression for energy. However, knowing the integral of motion makes easier finding solution of the equations of motion.
Similarly, Maxwell equations (that are equations of motion) cannot be derived from electromagnetic field stress-energy tensor.
On the other hand, the expression for stress -energy tensor can be used to derive expressions for some other important physical quantities.
If we consider compound system of charge + electromagnetic field, we can derive the Lorentz force acting on charge from the expression of EM field's stress-energy tensor and Maxwell equations:
$\frac{\partial T^k_i}{\partial x^k}=-\frac{1}{c}F_{il}J^l$
What is important here is that we use expression for the EM field stress-energy tensor and Maxwell equations (that are equations of motion for the EM field) to derive the force acting on charge.
But knowing the force acting on charge is not enough to derive equation of motion for the charge. We also need to know the "inertial properties" of the charge for that.
A: There certainly are systems that are fully described by their energy-momentum conservation as a simple example let us take one-dimensional particle in a potential. The energy is conserved:
$$ \frac{m \dot{x}^2}{2} + V(x) = E $$
Differentiating this expression w.r.t. time one gets
$$ \frac{2 m \dot{x} \ddot{x}}{2} + \frac{\partial V(x)}{\partial x} \dot{x} = 0 $$
$$  (m \ddot{x}+ \frac{\partial V(x)}{\partial x} )\dot{x} = 0 $$
This means that either E.O.M.'s are satisfied or the particle is at rest anywhere. If one can rule out the latter case by physical arguments, one arrives at a derivation.
The same is true for e.g. scalar field. However, as demonstrated in other reply (Sigma-model) -conservation laws might be spoiled in the way that they do not bring any information. This is fortunately not the case for classical fields.
A: The case of electrovacuum.
In General Relativity, from the Ricci tensor, we know the stress-energy tensor and vice-versa (by Einstein equation).
There are some conditions which, if are satisfied by the Ricci tensor (or equivalently, by the stress-energy tensor), are equivalent to the fact that the equations of motions are those from electromagnetism, and we can even determine the electromagnetic tensor (up to an overall "phase factor", or even uniquely in "charge without charge"). They are called the Rainich conditions, and were proposed in 1924-25 by G.Y. Rainich.
These conditions were rediscovered by Misner and Wheeler, as part of the charge-without-charge subprogram of geometrodynamics. For this, see Geometry of gravitation and electromagnetism, by L. Witten, and chapter 9 from Gravitation: An Introduction to Current Research, ed. L. Witten, and section 5.3 from Spinors and Space-time: Spinor and twistor methods in space-time geometry, R Penrose, W Rindler - 1986.
