Just to add to @antgonza's explanation — primer sequences should be trimmed prior to denoising with dada2, as explained here and noted in the dada2 faqs. In dada2, this can be accomplished easily with the trim-left parameter.
As far as I know, deblur does not have the same requirement.
Other downstream steps could still theoretically be affected. For example, if you train a feature classifier trimmed to a specific primer region, the trimmed reference sequences will not include that primer sequence. This probably does not have a significant impact on classification, but this is untested.
We have not really benchmarked these effects in any step because, as @antgonza pointed out, we (QIIME developers) are usually working with the EMP sequencing protocol, in which the primers are already trimmed from the reads.
I hope that helps!