Is it damaging to my voice to keep raising the key of songs to stretch my vocal and singing range

That is what a music or voice coach can teach you. What foes your vocal coach say? You did say you have one, right?

Just like any other over-used muscle, your diaphragm and/or throat muscles can tire.

It's probably not good. You can broaden your range with practice -- Placido Domingo, for instance, started as a baritone but made the transition to tenor very early in his vocal training. But there are limitations -- Domingo started out as a high baritone, and he never lost the darker coloring of his voice that true tenors, like Pavarotti, lacked.

If you do it right, you can add a few notes to the top of your range, maybe half an octave. On the other hand, if you do it wrong, you can ruin your voice permanently. If you have a vocal coach, follow his/her advice. If you're still straining and hurting, get a different coach. You should expect results, but don't expect miracles.

You should not make yourself hoarse. Vocal tiredness happens. The scales and vocal exercises done in moderation each day will stretch your vocal chords and increase your range. If you are hoarse then you are straining your voice and could eventually end up with nodes on your vocal chords. You do not want that!