Tanfield, who also crashed during the Tokyo Olympics, suffered a big crash after his bike went from under him as he attempted to keep up with Dan Bigham and Ethan Vernon.
A stretcher was brought out, but Tanfield was able to walk unassisted after receiving treatment. He could not continue, however, meaning the British team was not allowed a restart and was unable to finish.
"Following his crash in this morning's men's team pursuit qualification, Charlie Tanfield was assessed immediately by the on site medical team before being taken to hospital for further treatment," British Cycling said in a statement.
"We wish Charlie a very speedy recovery and look forward to seeing him back on his bike soon. Unfortunately, the men's team pursuit riders will not progress in the competition."
Niklas Larsen, Carl-Frederik Bevort, Frederik Madsen and Lasse Leth took Denmark to the top of the qualifiers with a time of 3:46.816, with New Zealand and Italy rounding out the top three.
The eight quickest teams qualified for round 1, with Australia, Canada, France, Germany and Japan also going through. Belgium, China, Spain, Switzerland and the United States failed to qualify.
Round 1 will be held on Friday, while the finals will take place on Saturday.
