The problem of designing a positive Luenberger observer for a class of infinite-dimensional semilinear positive systems is addressed. The proposed observer is positive. That is, it ensures that the state estimates are non-negative at any time. The positive observer is designed based on positive bounded perturbation results. In particular it is shown that, under suitable conditions the state estimation error converges exponentially to zero. The results are illustrated by numerical simulations based on an exothermic Plug-Flow Tubular reactor model and show the effectiveness of the proposed positive observer.