what is an example of a solution or reagent for which precision is very important?

Anything that involves explosives needs high precision for safety reasons.  If to much nitro glycerine is added to a compound for example, it could become unstable and deadly.

In the case of washing your car, it doesn't matter much if you use a tablespoon of soap per gallon or a quarter cup- not much will occur that would mess anything up.

Any solution for the purpose of injecting into humans or animals needs to be carefully prepared.

Examples:    saline solution,   potassium solutions,   glucose solutions,    anti-biotic solutions.

Almost any pharmacuetical solution,   eye drops,   ear drops,    heart medications,   any pharmacuetical student could add many many more.

You might want to use the word accuracy, rather than precision.  Excellent precision could mean you get the wrong answer every time and all your measurements are almost identical (but they are all wrong).

Accuracy is very important if you are manufacturing primary standards, such as a powder used to confirm the concentration of a titrant that will be used for titrations.

As an example, a powder with an accurately known concentration is used to determine the concentrqation of a sodium hydroxide solution that will be used for acid-base titrations.

For an example where we don't need it, hmmm, if we are mining salt to be used to melt ice on roads, we might want to have an idea of how much KCl and NaCl there is, but we won't have to know with as high a degree of accuracy.

I believe you are referring to precision of molarity, and will base my answer on that assumption
Precision is required where imprecision has a detrimental effect which is more costly than the effort made to obtain precision.

Important: Botox injections- the stuff is worth more than is weight in gold, definitely do not want to waste it or be sued by a patient for giving too little.

However everything needs a little precision to it, otherwise we would not be able to predict properties of the solution, making it useless.

Less important: Crystal growing kits- while the solute needs to be standardized, its molarity does not; a kid could overfill or under-fill by quite a bit, depending on the kit, while still getting a cool crystal in the end.

Reagent
Important: Potassium- the stuff controls your heart and nervous system.

Less Important: Home cooked bread- fluffy or flat the family will still appreciate it; not so for competition cooking.