Our bodies would radiate into electromagnetic energy like we were an antenna. Had a physics professor outline the equations in answer to this question about 40 years ago ... but after 40 years I have forgotten his development.
Yes, and our relativistic mass would approach infinity as we accelerated towards c as well. Since force=mass * acceleration, the amount of force required to further accelerate a mass that is approaching infinity would also approach infinity, which means you would require infinite energy to achieve that force.
Thus, no massive object can be accelerated to the speed of light, unless you somehow had an infinite energy source.