Am I burning bridges if I am applying for an internship which I am likely to turn down even if I am accepted?
Generally, you should not apply for a position you have no intention of accepting. I find it is unethical to do so; others may see it as merely impolite or even not a problem at all.
That doesn't seem to be the case here - you're applying and would take the offer if it's given to you and you don't have another better choice (better means better for you, not &quot;more prestigious institution&quot;), but might choose another offer if it's presented to you.
There's nothing ethically wrong with this action. Is it possible someone would take your rejection of an offer as a snub and react negatively to your applications in the future? Sure, there are people all over who make judgments on people for reasons they should not. However, that wouldn't be you burning a bridge, it would be someone else acting vindictively towards you. More likely no one will even remember if you apply again in the future.
Adding from Noah Snyder's comment, when you do get an offer you want to take, you should withdraw from other applications and decline other offers as soon as you have confirmed acceptance of the offer you prefer. It is rude to delay or remain in consideration for positions you will not take.