The case, which the Danish financial services authority called "very serious", is another blow to the reputation of Denmark's biggest bank which is trying to limit the fallout from its involvement in a major money laundering scandal.
Both the FSA and the public prosecutors "are pursuing the possibility of penalising the bank," the country's acting business minister Rasmus Jarlov said on Facebook.
The dismissal of Nielsen, who headed the bank's domestic banking activities and served as interim CEO until the end of last month, was the first major change since Chris Vogelzang took charge at the start of June.
"Every single day, we must strive to ensure that we offer our customers proper advice and products that suit their needs," said former ABN AMRO banker Vogelzang in a statement.
"As a result of misguided management decisions, we failed to do so in this specific matter, and for this we offer our apologies," he added.
Danske is separately under investigation in the United States and several other countries for payments totalling 200 billion euros (179 billion pounds) through its small Estonian branch, many of which the bank said were suspicious.
Danske will compensate around 87,000 customers who invested in the Flexinvest Fri product for a total of around 400 million Danish crowns ($61 million). Its shares traded 3% lower by 1240 GMT and have lost around a fifth of their value this year.
"Overall the charge is an unhelpful one-off at a time when public perception of the franchise remains in the spotlight," said KBW analysts in a note.
The fees for the product were raised in connection with the implementation of the MiFID II financial industry regulation in 2017.
"He (Nielsen) was at the time one of the responsible Executive Board members who did not to a sufficient degree ensure that the Flexinvest Fri product was suitable for the bank's customers," Chairman Karsten Dybvad said in a statement.
"Therefore, we find that Jesper cannot continue in his position," added Dybvad, who took up his role last year.
The Danish FSA said in had been investigating the case for some time and expected to make a decision on it by late summer.
"On the basis of the information available, this is a very serious matter," said Jesper Berg, who heads the Danish FSA.
Nielsen served as interim CEO from last October after former chief executive Thomas Borgen resigned when the scope of money laundering case emerged from an internal investigation.
(Reporting by Stine Jacobsen and Jacob Gronholt-Pedersen; Editing by Edmund Blair/Keith Weir)
By Stine Jacobsen