Jorgensen missed his side's fifth and final spot kick at the Nizhny Novgorod Stadium, allowing Ivan Rakitic to convert for Croatia to send them through to the quarter-finals.
After being subjected to abuse on social media, the Danish FA has filed an official police complaint and called on supporters to stop targeting the Feyenoord ace.
"STOP. Our society must never accept death threats - not against World Cup stars, politicians or anyone else," the governing body tweeted. "That is totally unacceptable and indecent. We are reporting the matter to the police to put an end to this madness."
Jorgensen has been touted as a summer transfer target for Premier League side Newcastle United.