Just ask your advisor in a very straightforward way if you should take it. E.g. "XYZ company asked me to do an internship with them, do you think I should take it or not". If he/she says yes then take it, and if not then don't.
If you have any concerns that your advisor might get upset at even being asked the question, then ask another student in your group for advice, e.g. a postdoc or a senior Ph. D. student.
There's really not much else we can say without knowing way more about your research area. You might be in a very collaborative field where your advisor is looking to form alliances and build relationships, and he might see this internship as a good way to build a partnership with this company. Or it might be a very competitive field where he is concerned about being "scooped", and would be afraid that you would accidentally give away some key knowledge. Or it could be somewhere in between. There's no way for us to tell which one you are in.
Edit: in response to your comment:
How to not burn bridges,
but also
it would be dishonest of me to ask for permission if I have already made up my mind.
These seem a bit contradictory to me. If you want to not burn bridges with your advisor, you may wish to consider framing it in terms of asking permission. If he says "No, I do not want you to take this internship, and if you do you cannot continue as my student", then you know for an absolute fact that taking this internship will burn a bridge with him. You don't have to guess if it will burn a bridge, you know. At that point, you still have a choice, you can burn the bridge or not. Now if you go in saying "I am taking this internship no matter what and I don't care what you think", well that might burn the bridge right there. On the other hand, if he says "yes", then you know there's no issue.
