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Even in the steady-state case, the statement that "there is no driving force for current changes" is false--there are still competing electrochemical gradients, opening sodium and calcium channels will still depolarize the cell, opening potassium and chloride channels will still hyperpolarize the cell. In fact, if we assume the presynaptic terminal is at 0 mV, then voltage-gated calcium channels will be open, calcium will flow into the terminal, and the synapse may release neurotransmitter even in the absence of action potentials.
Most excitatory synapses have a reversal potential around 0 mV. Since opening a synapse can only bring the cell toward the synapse's reversal potential, EPSPs would have no electrical effect on the cell in this case (although they would still have a chemical effect by allowing ions to flow down their chemical gradients). IPSPs would be much stronger. There is a very interesting question here--why do we go to the effort of hyperpolarizing the membrane at all? Why not simply have excitatory synapses with positive reversal potential (ie, only permeable to sodium or calcium)?
This is a bit misleading--a cell with 0 resting potential is not a 'dead battery'; it can still have chemical gradients and active synapses. I agree that action potentials will not be generated, but inhibitory synapses will still function (at least with spontaneous release) and excitatory synapses will still exchange ions when they are activated and admit calcium into the postsynaptic cell, even if they pass little net current.
