How can I make any mesh be filled with smoke?

If you're using Eevee, volumetrics nodes won't work as they work for Cycles, they will be shaped into cubes.

Since 2.91, though, you can create a volume empty (shiftA > Volume > Empty), and give it a Mesh to Volume modifier, with your object (here, your monkey) as Target. It will create a smoke that follows the shape of the Target:

Make your Suzane invisible. You can give your empty a material, by default it will be a Principled Volume plugged into the Volume socket of the Material Output. You can change the color or density of the volume:

Also, play with the parameters of the Render > Volumetrics panel, as it affects the quality of the volume: