Does a thermostat need to be connected to the fuse box? My builder says no?
There are two very different issues here:
Power to the Thermostat

I flipped off all the circuits in the fuse box, and the thermostat on the first floor powered off. However, the thermostat on the second floor still had power.

A thermostat is typically powered primarily by a transformer. However, many thermostats include battery backup. There are a variety of reasons, among them that unless a thermostat is wired up a particular way (see all the messages about Nest and C wire), it may only have power when the furnace or air conditioner is running. That is fine for an old mercury dial, but doesn't work for modern thermostats. Therefore, batteries. However, even with battery backup, a thermostat designed to always have power from the transformer will appear &quot;off&quot; when power goes out, using the battery just to save settings, clock, etc. A thermostat that is not designed to always have power will continue just fine when the power goes out - and even if pulled off the wall, using the battery to power everything.
Low Voltage Wiring/Devices

the electrician is claiming because it is low voltage, it does not need to go through the fuse box.

The electrician is correct about this, in a way. The great thing about most thermostats (there are some line-powered thermostats which are exceptions) is that they use low voltage and are therefore very safe to work with. While they indirectly go through the main electrical panel (by way of the transformer) the wiring actually present at the thermostat is considered separate from the electrical wiring at the main panel (the other side of the thermostat). That means it is reasonably safe to mess around with that wiring without the same level of precautions needed for 120V or 240V AC. The same is true for telephone and network wiring.