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 "more prestigious institution"), 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.

No

Offers are only offers, not contracts. They can be declined, and are declined all the time (for example, in 2018, 18% of undergraduates accepted by Harvard declined). Whoever is managing the internship will have a wait list, and if you decline, they will call someone else from the wait list and forget about you.

Naturally the more people are accepted in the internship the less likely someone is to remember you, but even if you're the only person, you should be OK especially if you decline courteously and promptly.

I think it pretty unlikely that such a decision would follow you into the future. Few would know about it unless you tell them (other than at that one place).

People understand applicants applying to several things at once, given the uncertainty of the process.

But you would do well to keep an open mind until you need to make some definite choice.