I came along the following question while trying to understand and apply some ideas of Dugger's article *Universal Homotopy Theories*. 


Suppose, we are given a nice model category $\mathcal{C}$, say left proper and cellular or combinatorial, so we have a good theory of localization. I am primarly thinking here of the category of presheaves of simplicial sets on some site with the projective model structure, where weak equivalences and fibrations are defined "pointwise". 


Suppose furthermore, $S$ is a class of morphisms in $\mathcal{C}$ we can (left Bousfield) localize at [e.g. the class required for descent for hypercovers] and $T$ is an arbitrary set of morphisms in $\mathcal{C}$. 


Now, let's consider a fibrant object $X$ in $\mathcal{C}[S^{-1}]$, i.e. some object which is $S$-local (and $\mathcal{C}$-fibrant), and take it's fibrant replacement $X^f$ in $\mathcal{C}[S^{-1}][T^{-1}]$. 


Is it now reasonably to expect under some circumstances or even generally true that the map $X \to X^f$ is a weak equivalence in $\mathcal{C}[T^{-1}]$?



Localizing Model Structures
(Your question is basically about presentable (∞,1)-categories, so I will take the liberty of writing my answer in that language. Hopefully the translations to model category language will be straightforward.)


Inside $\mathcal{C}$ we have the full subcategories of $S$-local objects, $T$-local objects, and $(S \cup T)$-local objects, which are all reflexive subcategories; denote the associated localizations on $\mathcal{C}$ by $L\_S$, $L\_T$, $L\_{S \cup T}$ respectively. Your question is whether for $X$ an $S$-local object, the map $X \to L\_{S \cup T}X$ is a $T$-local equivalence, i.e., whether $L\_T X \to L\_T L\_{S \cup T} X = L\_{S \cup T} X$ is an equivalence in $\mathcal{C}$. In other words, the question is whether, if I start with an $S$-local object $X$, the localization $L\_T X$ is still $S$-local.


In general this will not hold. For example, take $I$ to be the category $\ast \to \ast$ and let $\mathcal{C}$ be the diagram category $\mathrm{Fun}(I, \mathrm{Spaces})$. Write a typical object $X$ of $\mathcal{C}$ as $[X\_1 \to X\_2]$. There are sets $S$ and $T$ of morphisms such that the $S$-local objects are the ones for which the map $X\_1 \to X\_2$ is an equivalence and the $T$-local objects are the ones for which $X\_2$ is a point (I think they're $S = \{[\emptyset \to \ast] \to [\ast \to \ast]\}$ and $T = \{[\emptyset \to \emptyset] \to [\emptyset \to \ast]\}$). Then $L\_S[X\_1 \to X\_2] = [X\_2 \to X\_2]$ and $L\_T[X\_1 \to X\_2] = [X\_1 \to \ast]$. Clearly the $T$-localization of an $S$-local object need not be $S$-local.


One situation where I think the statement would hold arises from looking at models of a (finitary) essentially algebraic theory inside an ∞-topos. The idea is that the topos is a left exact localization of a category of presheaves of spaces, so that localization preserves the finite limits used to define the theory. However, that localization would need to be $T$, not $S$ as in your example, so I'm not sure whether this is the kind of example you had in mind.


