Tollen's Test and Werner's Theory
Ionizable and non-ionizable in this context refers to the observation that the counterions of a complex ionize in solution and can thus be precipitated and the ligands of the complex cannot and are thus non-ionizable.

There are two things at work here:


All complexes exist in an equilibrium with the free ligands. In the case of diamminesilver(I), $$\ce{Ag+ + 2NH3 &lt;=&gt; [Ag(NH3)2]^+}$$ the equilibrium is overwhelmingly toward the complex and no AgCl will be precipitated if chloride is added. However, if we add iodide instead, we will actually observe an AgI precipitate because AgI is far less soluble than AgCl and there is sufficient $\ce{Ag+}$ that we see some precipitate. Therefore, there is actually always a little bit of $\ce{Ag+}$ free in solution.
The Tollen's test is a redox reaction. Unlike the case above where we're precipitating an ionic compound and thus need to form an ionic bond, electron transfer to the silver is quite easy and once reduced, it is unfavourable for the silver to remain in solution. Even more extensive ligand coverage in something like $\ce{[Fe(CN)6]^{4-}}$ doesn't prevent redox reactions from occurring. Even some metalloporphyrins undergo redox reactions at the metal centre.