But what exactly will go wrong when it is stored at a higher
temperature or past its shelf life?
Solder paste is composed of metal microballs and flux. The metal doesn't change over time much; the flux does. Any solvent that maintains flux viscosity can 'dry up' over time and the total mixture becomes less viscous. This is a problem in a manufacturing environment because it can lead to improper flux distribution along a footprint during the stenciling process (especially if the flux (or lack thereof) balls up or has an affinity for itself more than what it is being dispensed onto).
This is mostly a problem in large PCB assembly lines. The problems is if you have any part of the process that doesn't function well, the total production yield goes down, and you have to spend time handworking your designs (like with a tombstoned SMT part). If you were producing quantities of 10000+ this could become a problem very fast, so you try to control every aspect of the design to increase the yield, and bad paste isn't something that assemblers want to deal with). Critical processes such as medical or aerospace will also require designers to follow expiration dates on solder.
That being said:
I operate a small line with quantities in the 10's per year for prototyping. I don't want to order flux every six months or whatever the expiration date is. I bought a large tube of no clean SAC305 solder paste, and it seems to do well well past the expiration date. I've had it for a few years and it still dispenses and spreads on the stencil\PCB well. I also keep this flux in as good of environment as possible by refrigerating the paste and keeping the cap on as much as possible
At a certain point, solder paste can become too dry and it gets hard to work with, but if you're working at home or a small line, all these problems can be handworked (we probably get a few more parts that don't solder correctly and we handwork those. It is much cheaper and faster than going to an assembly house for prototyping).
I have had to rework a few parts and add flux after the fact in a small number of cases, but if you can tolerate that, then use the flux for as long as it works for you. If you're in a production environment then follow the expiration date.