If I understand your question correctly, the AutoFields plugin should do the job for you. It "...allows you to configure vector fields to be automatically calculated for you when digitizing new features or modifying existing ones." Watch these videos for some demonstrations.
In your case, you would need to configure the existing
Height field (i.e., you don't need to create an additional field) as an AutoField with this expression:
"floors" * 3
Doing so, every time you change the building
floors attribute, the corresponding
height will be updated.