#1 - logically, how can a bad water valve cause the line to freeze? Something a competent tech should question even if it comes form incompetent mfg support.
#2 - if it is freezing inside the door, I would say there is a mfg defect in the door - like insulation pushing the water tube against the door instead of surrounding it.
If #2 is the cause, I don't think heating tape is going to help. If it is in warranty, use the warranty.
If it is older, and has not done this in the past, then it is anybody's guess.