Do the eye treatments with cold tea bags work?

Sure: Anything cold will reduce swelling.

A little bit: The caffeine in tea bags may produce a small amount of vasoconstriction which will help improve dark circles under the eyes. There are other topical products which will probably be more effective. For more severe bulging under the eyes, blepharoplasty is a effective long term solution. Seek a qualified facial plastic surgeon for specific recommendations.