If an Ising model is in contact with two thermal reservoirs, would it still experience a phase transition if one of the reservoirs is below Tc? For example;
Two reservoirs are at each end of a one dimensional or even two dimensional lattice.
One of the reservoirs has the temperature T < Tc. Would the lattice site have a phase transition even if the other reservoir has the temperature T > Tc?
Also as a follow up, is it possible to answer this question by starting with Glauber dynamics?
 A: Following up on Nathaniel's answer, a suggestion on how to achieve an Ising model coupled to heat baths only at the boundaries: one can allow in the bulk only moves that keep the energy constant, and only at the boundaries allow energy changing moves, with rates that correspond to different heat baths at the different boundaries. I'm not sure whether this would work with Glauber (spin flip) dynamics, but I believe it should work with Kawasaki (i.e., spin exchange) dynamics in the bulk, as this dynamics defines an ergodic Markov chain, i believe. Thinking about such a model in a lattice gas picture: there are particle hops in the bulk that conserve the energy, while at the boundaries patricles are injected from and removed to particle reservoirs with different chemical potentials and different temperatures. Another possibility (which differs from your question but is similar in spirit) is to have the system connected to a single heat bath everywhere, but connected as described to two different particle baths at the boundaries. 
Unlike equilibrium, this description does not fully determine the steady state distribution: not only the ratio of rates (which is determined by the reservoirs) is important, but also their actual values. This freedom can be used to our advantage: we can have the rates of injection and removal of particles and energy at the boundaries be very slow (i.e., scale inversely proportional to the system size to some large enough power). In this limit there is a timescale seperation between equilbration with a given energy/particle number and energy/particle non-conserving dynamics. Taking advantage of this timescale seperation, it is possible to calculate the steady state. In a one dimensional systems there would be no phase transition. In two dimensions, I believe that if one reservoir is below Tc, there will be a phase transition as one changes the temperature of the other reservoir. 
It is reasonable that this phase transition survives if the boundary rates are slightly increased, but as far as I know very little is known about the full nonequilibrium steady state when the boundary rates are not small as suggested. 
A: I guess in your case parts of the lattice will be at different temperatures (there will be a gradient of temperature), and local temperature will define the local phase of the lattice. As for @CuriousOne's comment, indeed, strictly speaking, a phase transition can only occur in an infinite system, but a large finite system's behavior can be very close to that of an infinite system.
A: I've played around with this type of question quite a bit. 
Your question leaves it a little undefined what you mean by the reservoirs being "at each end of [the] lattice." I will assume what you mean is that the lattice is divided into two sections, with one section being at one temperature and the other being at a different temperature, with no region "in between". (I suspect that what you actually want is for there to be an "in between" region that is not connected to either heat bath, but this is not straightforward to define in an Ising type model, since the assumption that every point is connected to the heat bath is a fundamental assumption in the model.)
If you do this, what will happen is quite simple. Each of the two regions will behave like an Ising model at its own temperature, and unless one of them is at the critical temperature there will not be any long-range interaction between the two. So in a two-dimensional system you will simply have two regions that each have their own phase transition independently of one another, and on a one-dimensional lattice you will have no phase transitions, because the Ising model does not have a phase transition in one dimension.
The reason this behaviour is so trivial is that the system has no conservation law. You're heating one end up and cooling the other end down, but the system has no way to store heat internally and transport it from the hot to the cold end, so all it can really do is go to equilibrium locally. There are various ways to go about changing this. One way to do it is to start with Glauber dynamics, but then add an additional rule that flipping a spin is much more likely if doing so will not change the total energy. If you do this it makes it possible for energy to be transported across the lattice without immediately equilibrating with the heat bath, and so the system can transfer heat from one part of the system to another. In principle, this sort of thing might be interesting, though in trying it myself I've never seen it do anything all that non-trivial.
I would say that the easiest way to get an intuition for this type of model is to simulate it yourself.
