It depends, and I don't know exactly which is the case in the MERA. The typical MERA does lose information. On the other hand, the MERA is best suited for describing self-similar systems, where the renormalization typically need not lose information, so I don't know whether there is a MERA that does not lose information. Looking at Swingle's http://arxiv.org/abs/0905.1317, he writes on p5: "The goal is to reach the ultraviolet by following the renormalization group flow backwards. This is possible because we have recorded the entire renormalization “history” of the state in the network, but subtleties remain because of the possible loss of information. In practice, the truncation error may be quite small with the proper use of disentanglers. More properly, the tensor network defines a large variational class of states for which the entanglement entropy can be computed by reversing the flow ".