The rlog transformation produces a similar variance stabilizing effect
as varianceStabilizingTransformation, though rlog is more robust in
the case when the size factors vary widely. The transformation is
useful when checking for outliers or as input for machine learning
techniques such as clustering or linear discriminant analysis.
For heatmaps and other downstream such as PCA or any kind of classification/machine learning one commonly uses vst/rlog or something like the normalized counts on the log2-scale. For Pearson correlation (cor) it depends what you want to show. The linear cor changes obviously when you apply a log transformation as log scale is not linear (which is the whole point of logs). If you want to see how your samples compare in terms of a traditional Pearson correlation then I'd use the raw counts or the output of counts(dds,normalized=TRUE) which will give the same correlations as they are on the same linear scale and normalization by DESeq2 is just dividing the raw counts by a single factor.
from DESeq2 help on rlog