If anything it may just need to reconverge. About 45 seconds of interruption IIRC. If you can, while you are changing it on all the switches anyway, use rapid PVST. That will allow for faster convergence.
I recently went through a similar migration that eradicated proprietary protocols (i.e., PVST+, HSRP, etc.); our transition was focused on posturing us for a 'vendor neutral' network. Since you're running Cisco devices, you're likely running PVST+.
Cisco covers this in more detail, but the crux of it is fairly simple: establish a hierarchy and, contrary to what common sense would tell you, start from the core and move outward.
Set your priorities in a manner that cascades down. Base this hierarchy on what is closest to a central node. All nodes participating in this STP setup should be on a common structure, so for simplicities sake, manually set one root bridge with the spanning-tree vlan xx,xx root primary.
Once you're ready for the migration to begin, start with the core. The reason for this is because MST is backwards compatible with PVST, not the other way around. A PVST core won't be able to communicate with BPDUs from MST nodes; those trunk links will drop. If everything is going right, you'll see a small drop while the new STP protocol comes online and figures out the current lay of the land.
Then just trickle these protocol changes down all the way to your furthest nodes.
End-users might not even notice the small blip while MST elects a root bridge.
Best Answer
(R)PVST(+) is a proprietary standard defined by Cisco (or a set thereof). However, there are many other vendors and devices supporting it - you may need to check the specifications of your devices.
The IEEE standard alternative is MSTP which enables separate spanning trees for different MSTP instances. Additionally, MSTP also supports splitting a network into separate regions or domains. MSTP is more complex but also more powerful than PVST.