The tax office in your home (EU) country will most likely ask for your (new) address in the country you're moving to before they approve your request.
I'm not saying this can't happen in the future... however if you are an EU citizen, you do not need an "approval" to move out or request something. You were born in some country and in case you want to leave (for whatever reason), the state doesn't have a limitless right to know where you will live, what will your address be etc. There are some exceptions (possibly Finland, France) where it is more complicated and you may get into trouble later if you don't actually report beforehand.
* You violate terms of the bank
* CRS, your bank in EU reports yearly to their state
CRS is primarily intended for cross-border reporting. If you are e.g. citizen of Germany and have German bank account, CRS doesn't apply; other reporting may apply though... It is true that he may violate the terms of the bank - so probably the worst that can happen from the bank's side is account closure.
To answer OPs question whether this can cause problems - yes. If you have accounts opened in your home country and you are actively using them (e.g. salary, credit cards...) it can be used against you to claim you have your center of vital interests still in your home country. If you just keep the accounts opened and won't use them, you can defend yourself (if that situation arose) that you just forgot to close them and you forgot to update your new residential address because you didn't know you still had that account in your home country. If you want to be 100 % certain, just close those accounts.
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Also I disagree with the fact that not telling your state where you live means you have something to hide. If you are a citizen, there really are no benefits if the state knows where you live - it can just be used against you (surveillance etc.). I'm from a country where everyone just believes this dogma that you MUST have an ID card and you MUST have a residential address and it is illegal not to have them. Actually when you do the research and study the law, you'll find out neither is mandatory.
In some countries ID card (and residential address) is not mandatory... for example Mexico or Sweden.
Some countries even do not have ID cards - Canada, United Kingdom.