this is true in the topological setting. cofibrations can be thought of as colimits, they are actually colimits of a diagram (i have been told, but i cant recall the example) and colimits commute with colimits. In the setting i am thinking of cofibrations are the distinguished triangles. So i doubt you will find a counterexample in general since the result is definitely true for at least one triangulated category. Unfortunately, i do not see how this could be extended to other triangulated categories.
It does not seem like a result that would be true in general given my above reasons for believing the result. I will ask about the diagram it is the colimit/homotopy colimit of.
Is there a particular triangulated category you are interested in?
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