Yes, they can – when things are done correctly.
Unfortunately, many homeowners with emergency generators were literally left in the cold during the recent storm. Modern furnaces (and heat pumps) utilize sophisticated electronics and require a steady source of “clean power”. In many cases homeowners use emergency generators which are not properly sized for the intended load and/or are not correctly installed. As a result the emergency power provided fluctuates (voltage and/or frequency) dramatically and causes furnaces to shut down. Many times the furnace will “trip off” on a power quality safety and then attempt to restart after a time out. This gives the appearance the furnace is malfunctioning even though it is operating – or at least attempting to operate – correctly.
Emergency generators can and will work well when properly designed, installed and maintained. Professional expertise is required to avoid unnecessary frustration and safety risks.