should I have safety concern of this rusty fireplace gas burner?
It's difficult to assess the degree of rust, especially in the pipe feeding into the floor of the firebox.   Natural gas can have a certain amount of moisture in it, which may be contributing to (causing) the rusting.  If this is the main source of moisture it would be more pronounced on any vertical pipe run.

I would suggest you remove as much of the visible rust as possible.  This will help indicate how much the vertical run might have.  Also check that portion of the pipe that is exposed below the floor (you indicate there's a shut-off valve beneath).

If there's more rust than what a wire brush can remove in a few (several) swipes then a plumber should be called.  Otherwise, if the rust can be quickly removed, a mixture of 1 part dish detergent + 1 part water sprayed on the entire exposed pipe (while gas is on) will reveal leaks via ensuing bubbling.  It's best if one person turns on valve while another sprays pipe assembly.