Can grout for a mosaic project been tinted? Food coloring, etc.

you can purchase colored grout.

This article from a mosaic art retailer says you can use acrylic paint: https://blog.mosaicartsupply.com/how-to-color-grout/

If you can't find the color grout you want, try adding powdered tempra paint to the grout; it will be brighter but the color will soften as it dries. If the piece will be out-of-doors, in a moist/humid area or..., You may want to use a grout sealer, as well.

Buy colored grout

Yes, mix in acrylic paint.

as you ask about food color.. it would fade due to lack of any pigment in it so no food color would not work but yes have it tinted any color to match what you want at a paint store. SHERWIN WILLIAMS, BENJAMIN MOORE stores.... :) acrylic paint is excellent for this...NOT oil !