The airbag may "hesitate" to deploy if the occupant is unbuckled. Also, during the crash, a non-buckled-up occupant may lift her/his butt off of the chair, on the way to her/his/its sure death, giving the weight sensor an impression the seat is not occupied.
From my understanding of air bags and belts, including the child-protection weight-sensor issue, I think it's very unlikely that the airbag deployment would be suppressed.
1. Three-point shoulder belts remain the single most effective and important tool in protecting the driver and front passenger. The original and still most basic reason for development (and later, mandated standard inclusion) of front airbags was that too many people were not being convinced to use shoulder belts in the 1970s through 1990s. This despite various ad campaigns and failed attempts to facilitate and/or force their use via clever auto-harnessing mechanisms, reminder lights & buzzers, and starter interlocks. Over time, the public has largely accepted them
2. In the absence of shoulder belts, the front airbag becomes hugely important as the primary life-saver and injury mitigator in the most commonly fatal front collisions. (Advanced multi-airbag systems, seat mounts, pre-tensioners and other features are notable but beyond the scope of this post.) I'd be extremely surprised and dismayed to be told that an unbuckled belt would de-activate the airbag deployment, for an occupied passenger seat or for the driver seat at all - which should never be unoccupied short of future L4/L5 scenarios! An unbuckled driver is exactly the most pertinent scenario needing airbag protection.
3. OTOH it is true that the energy of the airbag deployment, a very violent and itself somewhat dangerous event, may be adjusted downwards for children and lighter-weight passengers, and possibly also for small, lighter-weight adult drivers. This refinement occurred, I think, 10 to 20 years ago following some injuries and deaths blamed on the airbag deployment itself. But again I don't see a sensible justification for complete de-activation on the driver side
especially with an unbuckled belt.
I don't really love airbags myself, I love shoulder belts and intelligent crash-mitigation engineering of all kinds. I'm somewhat nervous when working around the dash in an airbag car. But truthfully airbag systems are getting a lot better lately.