Nice idea, but I don't think it works quite how folk have interpreted it here.
The GPS in the phone already returns a z-axis position (that's the altitude in the wgs84 coordinate system).
There's no problem at all pushing the data from the phone, it's absolutely commonplace these days.
A similar technique is already in regular use to do with travel times. The time taken to get from one place to another (in built up areas) changes as the amount of traffic changes. Several journey time app's use the users' GPS in the background, checking their speed, then using this to adjust the network model to current conditions.
It would be very easy and cheap to use this technology to build a very detailed and real-time pressure map.
Someone will figure out how to use this to work out the likely paths of tornado's in real time. That seems like an immediate win.
Edit. Re: using cell towers for location.
This is only a poor approximation of location... It can be done by triangulation between nearby cell towers IF the phone has contact with more than one at the time. But that requires the cooperation of the telco to achieve. Usually this technique only returns the location of the cell tower being used. Also, there's no z-axis value returned from this technique except that of the cell tower, which means my phone would report pressure where I am, but report it for the cell tower's location.
For those concerned about privacy, the app doesn't need to forward any user details to make this work, all that's needed is, lat, lon, alt, pressure and time.
The processing load isn't big in the scheme of things, but for performance reasons you'd probably seperate the data acquisition and management from the presentation and analysis.