Your Instagram doesn’t actually need to be instant
Wait to Instagram the exhibit until you’re in a phone-friendly zone, like a couch in the lobby. It’s pretty much impossible to experience an exhibit fully if you’re busy picking a filter and writing a caption. Waiting to post will also help you be more selective about what you share—which your followers will appreciate.
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Do not use the flash
If you’re taking a photo in a museum where photography is allowed, make sure your flash is off. Think of flash like a bolt of lightning that could destroy the material the artwork/artifact is made of. The museum will go to great lengths to insure this is communicated. I once went to an exhibition at the Met where there was a curtain over the work that needed to be gently lifted before viewing because even the dim gallery light was too much exposure for the precious piece.