Is there a good site or software to see if a primer pair spans an exon junction?
If there is a such a site, I have never encountered it. Since you say you have the genomic coordinates for each primer it seems to me like all you need is a list of the genomic coordinates of every transcribed exon in the human genome. Assuming that is correct then you are in luck because such lists do exist in the form of a GFF3 file. You should be able to obtain a comprehensive annotated GFF3 file of the human genome from the EnsEMBL website. It is possible that the UCSC browser has a similar file (except in their '.bed' format) but I  am less familiar with the contents of that site.