lmer() will typically omit any row of data that has an NA value for any variable that's in the model, outcome or predictor. From that perspective, if you don't impute then what other pre-modeling choice you make doesn't matter--rows with any missing data in the variables of interest simply won't be used in building your model. (As @Björn notes in another answer, other modeling approaches can incorporate missing data directly.)
The question then becomes whether this matters. There are circumstances in which it's OK to delete cases with missing outcome data, as explained by Stef Van Buuren where $X$ represents predictors and $Y$ outcome:
If the missing data occur in $Y$ only, complete-case analysis and multiple imputation are equivalent...
He notes two additional special cases:
The first special case occurs if the probability to be missing does not depend on $Y$. Under the assumption that the complete-data model is correct, the regression coefficients are free of bias... This holds for any type of regression analysis, and for missing data in both $Y$ and $X$.
The second special case holds only if the complete data model is logistic regression. Suppose that the missing data are confined to either a dichotomous $Y$ or to $X$, but not to both. Assuming that the model is correctly specified, the regression coefficients (except the intercept) from the complete-case analysis are unbiased if the probability to be missing depends only on $Y$ and not on $X$.
At a minimum, application of listwise deletion should be a conscious decision of the analyst, and should preferably be accompanied by an explicit statement that the missing data fit in one of the three categories described above.
Otherwise, multiple imputation is preferred, as illustrated nicely by @dipetkov on a data set you previously provided.
van Buuren's Flexible Imputation of Missing Data is a superb, generally accessible discussion of missing-data issues and how to deal with them, whether in your own data or in considering work by others.
In response to edited question:
I'm not fluent in tidyverse, but I infer that your specific question (if you aren't going to do imputation) is whether you should completely remove a Participant that has any missing values in modeled variables at any observation time in a longitudinal study.
A Participant can still provide information about observation times for which she provides data, as Björn's answer indicates. Mixed models can even handle completely different observations times among individuals, particularly if you model time smoothly (e.g., with regression splines). So the most general answer is: don't throw away data, keep all the useful data that you have.
Or use multiple imputation if appropriate.