Burrow, 20, sustained the problem when making a block tackle in the 4-1 victory over Barnet last weekend.
A scan revealed the damage to be more severe than first feared, with the player now set for a spell of 10 to 12 months on the sidelines.
Boss Jim Bentley told Morecambe's official website: "It's a huge blow for Jordan and for us.
"We didn't realise how bad it was at first but then we got the results of the scan and it was worse than expected.
"It is another blow for us as he is the second striker now ruled out for a time with the same injury and it leaves us a bit short of options but we will continue to work hard and assess all our options."