tl;dr your analysis removes missing data. You lose a bit of power, but the analysis remains unbiased when model assumptions are met. If an assumption is violated, there's not much a missing data method can do for you.
LMER in R will remove missing observations, i.e. a complete case analysis is performed. As with complete case analysis in a likelihood based procedure, depending on the nature of missingness, the resulting analysis is not biased, but is slightly inefficient.
I'm defining a cluster to be repeated measures within subject. Imbalance in cluster size would matter if you used methods for independent data. The resulting analysis would be anti-conservative meaning your 95% CIs would be too narrow to accurately describe the uncertainty. This is because those later observations have a larger residual correlation due to the subject-level uncertainty that is only correctly measured with a mixed model. The mixed model uses a random intercept; this imposes an exchangeable correlation structure meaning repeated measures within subject are positively correlated. The impact of this positive correlation is that repeated measures within subjects are relatively downweighted. This will adjust your power downwards for detection trends in time. Consequently the time by group interaction will be appropriately conservative if model assumptions are not met.
If the power is too low to perform a final analysis, some power may be recovered with integrated likelihood methods, i.e. expectation maximization (EM) or multiple imputation (MI) methods. Accounting for missingness with EM is more tractable than MI, but both are theoretically possible only on the basis that it would improve the efficiency of the analysis. They are complicated to do, require many assumptions, and would raise the eyebrows of a few reviewers.
To underscore all of this, the matter of bias creeps in only when we consider a rather grave type of violation. A model assumption of acute interest is informative loss-to-follow-up. Subjects may drop out early when
Group doesn't have the desired effect. This is an issue that may cause subjects contributing more measurements per time to "exaggerate" their specific effects as one which is population-level. The problem with this is that there really is no way to handle this.
One may argue for pessimistic types of imputation such as last observation carried forward (LOCF) or worst observation carried forward (WOCF), each presuming there was at least one negative observation prior to a subject's withdrawal from study.