Electrical advice concerning a 220v dryer outlet and circuit
You have more problems than just connecting up a dryer. Buy the way in answer to your question. NO you can't tap a range feeder and supply a dryer with it.

First you say your range feeder is protected by cartridge fuses. That would mean that you have and old Edison Base Fuse Panel or worse. In this state the AHJ recommend that when we as electrical contractors go into a dwelling it is our job to recommend the owner replace them.

Second NEC Article 220.55 requires a separate circuit for household ranges. The size is determined by the nameplate rating of the range. Since we don't know what size range the homeowner is going to install, we usually run a 50A circuit which covers all ranges up to 12KW. The minimum circuit it requires is a #8 wire and an over current protection of 40A. So you need to replace the 30A circuit you currently have.

Third NEC Article 220.54 requires a separate circuit to be run for the dryer. and It needs to be able to handle 5KW or the nameplate rating whichever is larger. So a minimum circuit would be a 30A circuit.

So if you have the capability in you panel to install a 50A circuit for your range. I suppose you could run a new range circuit and use the old range circuit for the dryer and relabel your panel to reflect the change.

Hope this helps and good luck.