It is free to close ordinary deposit accounts, if you have had the account for more than 12 months, but your old bank may require payment in certain cases i.e. to liquidate your loan or transfer your pension savings. Your new bank may require payment to set up a new overdraft, loan and pension savings. Ask if you are in doubt.
If you wish to move your loan, your new bank will carry out a credit rating and your new adviser will typically ask about your job situation and ask for your most recent wage slips, annual statement from SKAT (the Danish Customs and Tax Administration) etc. Your new bank will need to get to know your financial situation to be able to give you the best advice.