I have a tenant on the 1st floor who controls the thermostat. It's constantly on 80 and the 2nd floor is dying from the heat. Can I install a 2nd thermostat in the basement and limit the heat output?
If you install two thermostats in series, they'll both have to be calling for heat for the heating to come on.
For example if thermostat 1 is set to 80, and thermostat 2 is set to 75. When the temperature drops below 78, thermostat 1 will close and call for heat. However, the heating will not come on. When the temperature drops below 73, thermostat 2 will close and call for heat. At this point the heating will come on, and will stay on until thermostat 2 is satisfied.
In a situation like this, thermostat 1 will never be satisfied, and will always be calling for heat. You'll basically be bypassing thermostat 1. So the temperature near thermostat 2 will be comfortable, while the temperature near thermostat 1 will likely be somewhere less than 80.
Keep in mind, however, if the user of thermostat 1 figures out what you've done. They could turn their thermostat way down, and the heating would never come on.
The best (but not cheap) answer would be to make the two units separate heating zones, installing dampers (if forced-hot-air) or additional pumps and loops (if forced-hot-water) to direct heat only to the thermostat which is calling for it. That would give you full independent control within each space. Nothing else is likely to do so, I think.
You can install two sensors and remove the thermostat to an area of your control. The sensors are in series to the one thermostat.
Assuming they are basic, analog/mechanical thermostats then yes, you can wire as many in series as you want. Heating will continue until the room with the lowest-set thermostat reaches that temperature.
However, you must consider the whole-building heating process as just putting a 'stat in a random location could easily make the problem worse. It sounds like putting one in the second floor would be a better solution.