Does using furnace increase smithing skill?

No

Using a Smelter does not increase your skill, similar to a Tanning Rack.

Only actually making something using the Forge or improving things using the Grindstone or Workbench will give you Smithing experience.

NO, but, it could if you wanted it to...

As alexanderpas said, no in the default game, however, if you're playing on a system that supports mods, and if you're interested in such things, the mod Ars Metallica by Arthmoor:

• Provide skill advancement for the use of the smelter to melt down ore. After all, that's as important as knowing how to shape it once refined.
• Provide skill advancement for the use of the tanning rack to make leather. That's far more important in my mind than the skill advances from using the leather to make stuff.
• Provide for a small amount of skill advancement from mining the ore itself out of ore veins.

This mod is available on Nexus by Arthmoor for both oldrim and special edition and on BethesdaNet also by Arthmoor for PC, XBox One, AND for PS4.

Two things of note: Arthmoor does a great job with mods and he's very prolific so definitely check out his stuff. Personally I don't care for this mod, but that's really just me (as noted by the high rating of this mod everywhere).