We introduce a long-memory dynamic Tobit model, defining it as a censored version of a fractionally-integrated Gaussian ARMA model, which may include seasonal components and/or additional regression variables. Parameter estimation for such a model using standard techniques is typically infeasible, since the model is not Markovian, cannot be expressed in a finite-dimensional state-space form, and includes censored observations. Furthermore, the long-memory property renders a standard Gibbs sampling scheme impractical. Therefore we introduce a new Markov chain Monte Carlo sampling scheme, which is orders of magnitude more efficient than the standard Gibbs sampler. The method is inherently capable of handling missing observations. In case studies, the model is fit to two time series: one consisting of volumes of requests to a hard disk over time, and the other consisting of hourly rainfall measurements in Edinburgh over a two-year period. The resulting posterior distributions for the fractional differencing parameter demonstrate, for these two time series, the importance of the long-memory structure in the models.