Should I tape/support drywall seam next to cabinet?
It's unfortunate that you didn't leave a bit of drywall below the cabinet. If you had, you could then either tape the new to the existing or float backing behind the joint, screwing through the upper and lower panels. 

As it is, this is entirely a matter of opinion. Do you need backing? No. Your wall will not fall apart without it. However, it'll be much less rigid at the top edge, potentially allowing for problems with your tile. Movement cracks grout (and potentially the tile itself). If anything presses firmly or bumps between studs, damage could occur. 

So the question becomes about how can you put backing in there without undue destruction of your vapor barrier and degradation of your insulation, should you deem it important. I don't have any brilliant ideas aside from toenailing one-by lumber in place behind the poly. 

You could also use a rigid enough backer to avoid the whole issue. 5/8" plywood or 1/2" cement board might do the job.