Denmark’s fighter defence was to small to provide any resistance against the German invasion on 9 April 1940 and no fighter made any claims (at least not to my knowledge) during this short operation. However, at least seven Danish pilots fought with the Finns against the Soviet Union in the Winter War. Three of the pilots made biplane claims during this conflict and they were:

A fourth biplane fighter pilot was Luutnantti Carl M. H. Kristensen of LLv 26, who was shot down and killed in Gloster Gladiator GL-259 on 29 February 1940. Please see Poul Christensen for little more info.

These four pilots flew biplanes in the Finnish Air Force. At least three other Danish pilots flew for the Finnish Air Force during the Winter War, but these pilots flew monoplanes and were:

Luutnantti Mogens F. Fensboe
He joined the Finnish Air Force on 22 December 1939 and was posted to LLv28, which was to be equipped with Morane Saulnier MS.406 fighters, received from France. He flew in combat in the later stages of the conflict and managed to claim one victory, an I-153 over Viipurinlahti at 16:30 on 9 March 1940 while flying MS-320.

Luutnantti Erhard K. J. Frijs
Frijs flew with LLv24 on Fokker D.XXI fighters. He claimed a total of 1 and 1 shared victories and 1 damaged. The first was a SB from 50 SBAP over Virolahti in FR-114 at 13:25 on 30 January 1940, claimed as a shared victory together with kapteeni J. Carlssson (FR-91) and lentomestari Y. Turkka (FR-83). The second was a SB of 31 or 54 SBAP on 17 February 1940 in FR-100 over Kämäränjärvi. In the same combat, he claimed a second as a damaged. Both these claims were made at 09:20.
Frijs was killed when I-16s from 25 IAP shot down his Fokker (FR-80) over Käkisalmi (Kannas area) on 19 February 1940.

Luutnantti Fritz Rasmussen
He joined FAF on 26 December 1939. He flew with LLv 24.
He was shot down and killed on 2 February 1940 by an I-16 from 25 IAP over Rauha in Fokker FR-81.

Sources:
Lentolaivue 24 - Kari Stenman and Kalevi Keskinen, 2001 Osprey Publishing, Oxford, ISBN 1-84176-262-8
Most of the information here have been kindly provided by Tuomo Lukkari.
Additional information kindly provided by Jouko Rautio.