I think you should still be able to go over top depending on what you are using for flooring. And it also depends if it is a sheet of lino or those 12 x 12 squares.
If you're using laminate or hardwood flooring the missing lino shouldn't be a big deal. Long boards will likely cover those spots. If you want, get some Plani-patch to level and fill the missing lino areas....
http://www.mapei.com/public/COM/prod...tch_TDS_EA.pdf
Or, what would be best, is to install a 3/8 plywood subfloor over top of the existing lino. Of course it adds height to the floor but if you can make it work that is the best. Laminate would be easy to install and tile might be ok if you have a total subfloor of 1 1/8 inches.
The condo board will probably require special sound proof underlayment if you use hardwood or laminate to be aware of that.
If you want to tile and you have sufficient subflooring and it is sheet lino, you could go over top of the lino with a layer of Ditra. It is only 1/8 think so doesn't add much height. You have to be certain to use the right mortar to attach it though. I think it would be the unmodified mortar for between two non porous surfaces.
I'd probably not bother with removal and remediation.