Can spike protein induced cell fusion be triggered by the mRNA vaccine?
The mRNA vaccines encode a mutant version of the spike protein in which the structural transition needed to to fuse membranes is blocked. This was done to make the immune response focus on the pre-fusion state, which is much better for neutralizing the virus.
From https://cen.acs.org/pharmaceuticals/vaccines/tiny-tweak-behind-COVID-19/98/i38:

Fortuitously, Graham and a former postdoc, Jason McLellan, devised a solution to this problem before the pandemic. Through a bit of structural biology and persistent protein engineering, McLellan discovered that adding two prolines—the most rigid of the 20 amino acids—to a key joint of a vaccine’s spike protein could stabilize the structure’s prefusion shape. This 2P mutation worked in preclinical studies of Graham and Moderna’s MERS vaccine, so they applied it to Moderna’s COVID-19 vaccine.

I don't know if cells expressing wild-type spike protein would fuse into syncytia, but it seems very unlikely to happen with the vaccine spike.