When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations? Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on 
$\mathcal{C}.$ 
Assume that the model structure on $\mathcal{C}$ lifts to a model structure on 
the category of $\mathrm{T}$-algebras $\mathrm{Alg}_{\mathrm{T}  }(\mathcal{C}),$
where the weak equivalences and fibrations are those of $\mathcal{C}.$ 
Assume that $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves homotopy colimits indexed by $\Delta^{\mathrm{op}}.$ 
Does the forgetful functor $\mathrm{Alg}_{\mathrm{T}  }(\mathcal{C}) \to \mathcal{C}$ preserve homotopy colimits indexed by $\Delta^{\mathrm{op}}?$ 
More generally one can ask the question replacing $\Delta^{\mathrm{op}}$ by an arbitrary category. 
Remark: 
If $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves 
colimits indexed by some category $\mathrm{K}, $ 
the forgetful functor $\mathrm{Alg}_{\mathrm{T}  }(\mathcal{C}) \to \mathcal{C}$ preserves colimits indexed by $\mathrm{K}.$ 
This also holds for $\infty$-categories with the appropriate notion of monad and algebras over a monad.
 A: In Theorem 1.6 in the paper Bar constructions and Quillen homology of modules over operads, John Harper gave a positive answer to your question when T is the monad of a one-colored symmetric operad O and the underlying category is either symmetric spectra or unbounded chain complexes over a field of characteristic zero.  It is not hard to extend his argument to cover the case of colored symmetric operads.  I would not be surprised if his argument actually works for simplicial monoidal model categories, but I have not checked it myself.
A: 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.
