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$\ce{CsCl}$ and $\ce{NH4Cl}$ have similar reactions. For example, they can react with $\ce{AgNO3}$ to produce $\ce{AgCl}$ precipitate.
So my question is:
Can we use a chemical test to distinguish between solid $\ce{CsCl}$ and $\ce{NH4Cl}$?
The $\ce{Cs+}$ ion is very unreactive and I do not know any reactions involving it. I believe the chemical test can somehow convert the $\ce{NH4+}$ ion into $\ce{NH3}$ gas so that we can test it with a moist red litmus paper.
Correct you can distinguish them by their reaction with a base. You should use a base, such as $\ce{NaOH}$. $\ce{Cs^+}$ wouldn't react with $\ce{NaOH}$, while $\ce{NH4+}$ will undergo reaction $$\ce{NH4+ + OH- -> NH3 + H2O}$$ $\ce{NH3}$ is a gas and can be detected with a moist litmus paper.
If you will, you can actually detect $\ce{Cs+}$ using a procedure from this paper:
"Selective Detection of Cs+ in Water Solutions via One-Step Formation of a New Type of Struvite-Like Phosphate". However, this is an exotic reaction (in practice you would use the flame color test) and you don't have to do it since you only need to distinguish samples.
If you observe the two salt, you can see that the cations are different. So, you will have to differentiate the cations. There are two test that can differentiate the cations.
$\ce{Cs+}$ gives blue-Violet color while $\ce{NH4+}$ don't give flame test. For more information about flame test, see Jan's answer to this question. It also gives some information about $\ce{NH4+}$ detection.
Nessler's reagent test for $\ce{NH4+}$ (from same question). For $\ce{Cs+}$, there is no specific test for its detection and only rely on flame test. But, if it is said to separate $\ce{Cs+}$ from other alkali metal ions, it is separated by noting the difference of solubility of its platinochlorides, acid tartrates and alum (source) or estimated gravimetrically (source).
Just dissolve some of the stuff in water and apply blue litmus paper or pH paper. Ammonium chloride is weakly acidic. Cesium chloride is not.
Similarly magnesium hydroxide will be readily soluble in an ammonium chloride solution which is acidic, but not in a cesium chloride solution.
