Averaging can be attractive or convenient. It can also be a source of deception, at worst deceit, so tread carefully even when there is a clear rationale for averaging.
Here is a situation it which it is not a good idea. Consider that by careful definition of groups you (usually) could reduce your data to two summary points each distinct on the two variables; and then you would achieve a perfect correlation with magnitude $1$. Congratulations, or not! The improvement here is bogus without a good independent reason for the procedure. You don't need to approach this extreme case to approach the danger.
There are some situations in which averaging can make sense. For example, if seasonal variations are of little or no interest, then averaging into yearly values creates a reduced dataset in which you can focus on those yearly values.
In various fields, researchers could be interested in correlations at quite different scales, e.g. between unemployment and crime for individuals, counties, states, countries (substitute whatever terms make most sense).
The interest, and often also a major source of inference troubles, is in interpreting what is going on at different scales or levels. For example, a high correlation between unemployment rate and crime rate for areas doesn't necessarily mean that the unemployed have a higher tendency to be criminals; you need data on individuals to be clear on that. Data provision can be maximally awkward in data being available only on the least interesting scale, perhaps as a matter of economy or confidentiality.
I note also that that many measurements are in the first place often averages over small time intervals and/or small space intervals, so the data often arrive averaged in any case.