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I looked at the chrome inspector to see what's going on, and I noticed an iframe is being used to display the tags (below the note title, and above the options bar). This is pretty strange behavior, but I'm sure you have a good reason for it.
For that reason, it would likely be very difficult for you to make the iframe automatically expand in height to its contents because I have tried to do this in the past and have had varying success with it.
So, unless you ditch the iframe, I think the only way you can solve this problem is to disable word wrap for tags, and then show the overflow in a popup menu, similar to how Evernote does it.
I think a solution would be to display the tags in a div instead of an iframe, but I have no idea why you're using an iframe in the first place--so, I don't really know if that's a solution.
Quick Tags (autocomplete) is an extension, and extensions typically run in iframes. We have plans for re-evaluating this architecture in the future, but it will likely be a massive undertaking. There are indeed issues with wrapping with this particular extension, for which there are no good solutions.