My DH and I were faced with the exact same situation when we were renovating our old house.
What we ended up doing was buying a Johnson wood cleaner that warned DO NOT USE ON ASPHALT TILE. We would pour that stuff on the wood/linoleum edge and let it soak in and work it under the linoleum for a bit, and then pop off the tile and scrap off the tar with a big putty knife. We went through lots of cleaner and rolls and rolls of paper towels, but the floor was not marred at all when we got done.
You just need to be careful of the waste products so you don't get spontaneous combustion and burn your house down.
Many thanks for that info! I always wonder what the heck these people were thinking, using tar on original hardwood. But like a friend of mine once said, “Some people just ain’t very smart”. How right he was.