Identification of lithium and strontium salts
If your solution contains only $\ce{Li+}$ or $\ce{Sr^2+}$, you can differentiate them by adding a sulfate solution (for example, $\ce{Na2SO4}$ or diluted sulfuric acid). If strontium is present, a white, fine-crystalline precipitate of $\ce{SrSO4}$ will form. In the case of lithium, the solution will remain clear. This is because $\ce{SrSO4}$ is much less soluble in water than $\ce{Li2SO4}$ ($\pu{0.0132 g}$ versus $\pu{34.8 g}$ in $\pu{100 g}$ of $\ce{H2O}$ at $\pu{20 ^\circ C}$ (reference)). 

Adding diluted $\ce{HCl_{aq}}$ and $\ce{BaCl2}$ to a solution is commonly used as a test for $\ce{SO4^2-}$ (if it is present, $\ce{BaSO4}$ precipitates). Maybe this could help you to distinguish between $\ce{Li+}$ and $\ce{Sr^2+}$ given the different solubilities of their sulfates. If your unknown salt readily dissolves in water and $\ce{SO4^2-}$ is found to be present in the solution, then it is likely that you have lithium sulfate. Strontium sulfate as unknown salt would not readily dissolve and therefore not give a positive test.

However, if your solution can contain more ions besides $\ce{Li+}$ or $\ce{Sr^2+}$, I would suggest to try the separation with ammonium carbonate which is already mentioned in the comments to your question.