If the functor T preserves cofibrant objects, then preservation of sifted homotopy colimits follows from the preservation of sifted colimits by the forgetful functor and the fact that sifted homotopy colimits can be computed by replacing the diagram by a weakly equivalent projectively cofibrant diagram.

T preserves cofibrant objects in many situations of interest, e.g., for simplicial sets or chain complexes this is automatic, in many other situations it is true because T is a cocontinuous functor with good properties, etc.

Denote by U: Alg_T(C)→C and Free: C→Alg_T(C) the adjoint functors between Alg_T(C) and C. Suppose U(j) is a cofibration in C for all j, where j is a cobase change of Free(i) in Alg_T(C), where i is a generating cofibration in C.

In this case the preservation of sifted homotopy colimits follows from the preservation of sifted colimits by the forgetful functor and the fact that sifted homotopy colimits can be computed by replacing the diagram by a weakly equivalent projectively cofibrant diagram.

This follows from the key fact that the forgetful functor from sifted diagrams of T-algebras in C to sifted diagrams in C preserves projectively cofibrant diagrams. Indeed, the forgetful functor preserves sifted colimits and sends cobase changes of free morphisms on generating projective cofibrations to projective cofibrations by assumption on T.

This condition is satisfied in many situations of interest, e.g., when T is induced by a colored operad in a symmetroidal model category, as explained in Theorem 6.6 of arXiv:1410.5675. The model categories of simplicial sets, simplicial symmetric spectra, and chain complexes in characteristic 0 are symmetroidal. If we replace symmetric operads with nonsymmetric operads, then a tractable monoidal model category will suffice, which includes almost all important examples.