It will always be your own responsibility to check up on the latest rules and regulations concerning tax issues.
But generally speaking and according to the Danish Customs and Tax Administration’s order dated 30 October 1991 grants received for exchange purposes are tax-free “when they are granted with the clause and purpose that they should be used for covering travel and accommodation costs/purposes”. It is therefore important that you provide documentation for all major expenditures during the exchange period, so that you can document to the Danish Tax authorities that your grant(s) actually has been used to cover expenses during your exchange. (So – always save your receipts – also boarding cards for airplanes etc.)
If you receive Danish SU – then check with the tax authorities regarding tax on SU and other scholarships during your exchange. In some cases you may avoid paying Danish income tax of your SU during an exchange.