I agree with the other answers that you are not burning bridges with the institution. However, you may -- in some circumstances -- be burning bridges with individuals.
If an internship (or fellowship) is one of several offered by a group or department, or is just based on accomplishment not on fit with specific work requirements, there should be no problem. Your slot will just go to whoever is next on the waiting list.
However, if you specifically were selected since (formally or informally) someone spoke up to express enthusiasm to supervise you or otherwise have you work on their specific project, they may be now emotionally invested in you coming (and feel a bit rejected if you turn it down), and/or may have prioritized you over others in a way they cannot roll back. It shouldn't be that way, but selection meetings can become contentious with people backing their preferred candidates and emphasizing their strengths over others.
People do reject offers quite frequently and -- if you do so in a timely and responsible manner -- this will not make you enemies. But you do need to consider the possibility that specific people who may have lobbied on your behalf may feel less enthusiastic to invest more personal energy on your behalf next time you pop up on their radar screen in whatever other context.
Therefore: go ahead and do what you have to do, but be sensitive to who may be individuals who may (appropriately or not) feel a bit slighted if you do take another opportunity. If there are such individuals, it would likely be helpful if you reach out to them afterwards, explaining how you had to prioritize such and such, and reiterating how much you hope to work with them in the future (or somesuch, whatever feels natural). A bit of empathy and explanation goes a long way!