Motivated by recent experiments, in which knots have been tied in individual biopolymer molecules, we use Langevin dynamics simulations to study the diffusion of a knot along a tensioned polymer chain. We find that the dependence of the knot diffusion coefficient on the tension can be non-monotonic. This behavior can be explained by the model, in which the motion of the knot involves cooperative displacement of a local knot region. At low tension, the overall viscous drag force that acts on the knot region is proportional to the number N of monomers that participate in the knot, which decreases as the tension is increased, leading to faster diffusion. At high tension the knot becomes tight and its dynamics are dominated by the chain's internal friction, which increases with the increasing tension, thereby slowing down the knot diffusion. This model is further supported by the observation that the knot diffusion coefficient measured across a set of different knot types is inversely proportional to N. We propose that the lack of tension dependence of the knot diffusion coefficients measured in recent experiments is due to the fact that the experimental values of the tension are close to the turnover between the high- and low-force regimes.