I've stumbled upon the following which I can not prove.
For topological spaces, the telescope $$hocolim(X_0 \rightarrow X_1 \rightarrow X_2 \rightarrow ...)$$ is homotopic to $$ hocolim(\bigvee X_i \stackrel{id \vee id}{\leftarrow} \bigvee X_i \vee \bigvee X_i \stackrel{id\vee \bigvee f_i}{\rightarrow} \bigvee X_i \vee \bigvee X_i) $$
If we were simply dealing with colimits, I think I would simply show that it satisfies the universal property. However, I am not sure how to tackle the problem when dealing with homotopy colimits.