In many predictive tasks, there are a large number of true predictors with weak signals, leading to substantial uncertainties in prediction outcomes. The polygenic risk score (PRS) is an example of such a scenario, where many genetic variants are used as predictors for complex traits, each contributing only a small amount of information. Although PRS has been a standard tool in genetic predictions, its uncertainty remains largely unexplored. In this paper, we aim to establish the asymptotic normality of PRS in high-dimensional predictions without sparsity constraints. We investigate the popular marginal and ridge-type estimators in PRS applications, developing central limit theorems for both individual-level predicted values (e.g., genetically predicted human height) and cohort-level prediction accuracy measures (e.g., overall predictive $R$-squared in the testing dataset). Our results demonstrate that ignoring the prediction-induced uncertainty can lead to substantial underestimation of the true variance of PRS-based estimators, which in turn may cause overconfidence in the accuracy of confidence intervals and hypothesis testing. These findings provide key insights omitted by existing first-order asymptotic studies of high-dimensional sparsity-free predictions, which often focus solely on the point limits of predictive risks. We develop novel and flexible second-order random matrix theory results to assess the asymptotic normality of functionals with a general covariance matrix, without assuming Gaussian distributions for the data. We evaluate our theoretical results through extensive numerical analyses using real data from the UK Biobank. Our analysis underscores the importance of incorporating uncertainty assessments at both the individual and cohort levels when applying and interpreting PRS.