that this lino is NOT in squares. If it was, you could remove the damaged squares in re install new ones, IF you saved some from the last install.
If sheet lino or vinyl, then I would suggest two things to consider. One is to clean out under the tear with alcohol and a q tip. Remove all the debris. Get a tube of liquid nails and with the q tip again apply a small amount of that on the underside. Now get a hair dryer and warm up the torn pieces and lighty heat them until they are real soft and pliable. Push the torn section down on the glue and find some weight to apply on the cut to hold it down. Make sure that you use some wax paper in case the glue comes up from the floor and does not stick the weight down.
Second idea, if this does not work, go to the home center and get some vinyl filler and patch the area.