It is likely that Instagram regenerates some URLs – I don’t know; I have never had any 403 error for reasons other than using JetPack image optimization. In that case a cached copy of Instagram’s URL would get rendered void after a period of time. I haven’t had users use caching plugins with Instagram, so this is conjecture.
But what I would like to know is how I can prevent this from occurring in the future. Or, when it occurs, how can I be automatically notified so I know to login and clear the page cache? If I hadn’t logged in to clear the page cache, how long would these images have remained broken?
If anything, these questions should be directed towards your caching plugin. How frequently does it monitor the cache it has generated for dead links? How does it notify you of such instances?
Think of this like a TV channel that broadcasts all performances of a play by a troupe in a theatre live. Now you wanted to optimize something, so you taped one performance. But at a later point the playwright made some changes to the play, and the lines are now different (maybe a character was written out and two others were introduced). So the TV channel live telecasts have the new changes … but your copy doesn’t. Would you ask the TV channel why it hasn’t notified you of your copy being out of date?
The case is similar here. Photonic works real-time; its whole premise is showing content dynamically from a different website even if that content is changing (i.e. if you add new photos etc. they should show up without your having to edit your pages). Think of it as a dynamic embed of an external link. To that end it is not the objective or intention of Photonic to monitor your site for dead links (particularly those in a caching plugin’s cache), as by itself Photonic will never generate dead links – all links shown on your page at a point of time are active links generated by Photonic right at that point of time. Are these links constant? Probably – for Flickr, SmugMug, Zenfolio and Instagram (until your post) I have never seen the links change. For Google I know that the links change every 30 minutes, so caching is pointless.
If you want to play it safe, I would advise against caching any page with dynamic external content (not just Photonic, but even if you are including something like a Twitter feed).