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In the many-body localized phase, the system is described by quasi-local integrals of motion ("l-bits"). The entanglement does grow logarithmically with time. So if I use TEBD to get the real-time evolution will it be efficient? Or it will not work at strong disorder?

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If the entanglement entropy scales like $\sim \log t$, then the required bond dimension (and hence, the computational cost) for TEBD scales as a power law in $t$ (because it's exponential in entanglement entropy). If you call that "efficient" then TEBD is efficient. But if you want to go to very late times you're obviously still going to have problems.

For example, in my paper https://arxiv.org/pdf/1603.08001.pdf, we used TEBD to simulate the time-evolution of an MBL system at short times. But we also needed to go to very late times (for example $t \sim 10^{10}$ in some natural units) and for that we had to resort to exact diagonalization.

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