Myujth

Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_n+1: X_n+1to X_n), (q_n+1: Y_n+1to Y_n), (f_n: X_nto Y_n), n=0, 1,ldots$ with $q_n+1f_n+1=f_np_n+1$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $lim f_n$ is also a weak equivalence in $L_CM$?

For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.

No. For a counterexample to your claim, consider the model category M
of simplicial presheaves on a small site S equipped with the projective
model structure.
Its fibrant objects are presheaves of Kan complexes.
If C is the set of Čech covers of S, then L_C(M) is the local projective
model structure on simplicial presheaves.
Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
is a homotopy sheafification map.
Furthermore, the limit of p and q is a homotopy limit in M,
so lim f_n is a weak equivalence if and only if the homotopy sheafification
functor preserves homotopy limits of towers.
This is false for arbitrary sites.

In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.

Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).