Is it safe for a hole to be left open in my tooth for internal bleaching?

The hole through with bleaching was done for the tooth should be closed and sealed well after every appointment. The inner dentin gets softer to the food products and acidic foods. 

As another doctor already has mentioned, once an access has been created behind the tooth, the bleaching pellet is placed inside the pulp chamber of the root canal treated tooth and sealed with a semi-permanent material so that it protects your soft tissue from any seepage of whitening compound or gel that has been placed inside the tooth. Leaving the access open will only lead to leakage of the whitening gel and irritation of the tissue. What you describe does not seem to be correct in terms of how the procedure should be done. Discuss your concerns with your treating doctor. Good luck!

Normally with internal bleaching, a pellet with a bleaching solution is place in the tooth and sealed with a temp filling. Not sure why they would leave it open and use a bleaching tray, but that is not standard for internal bleaching.

The hole should be covered with a temporary filling. Sometimes the filling is a little irregular and can feel like a hole to your tongue. If it's definitely not covered, speak to your dentist. Most likely they placed a temporary filling and it fell out or a piece broke off.

This is a relatively common procedure, and yes, it is fairly safe.  There should be a temporary covering or filling in the tooth while you are doing the internal bleaching or "walking bleach" as it has been called.  Once the bleaching is complete, a more definitive restoration can be placed, which seals better.  Just brush your teeth thoroughly everyday and you should be fine.