Let's see (US perspective on this issue):
A hand dryer probably uses around 3000W to run the motor and the heater. It might be on for about 30 seconds. During that time, it consumes 1.5 kWm (kilowatt-minutes) or about 25 watt-hours.
According to this source, about 45% of the US energy supply (the largest proportion) comes from coal as of 2010. (185 GWh / 412 GWh = 44.9%), so I'll pretend that the energy comes from coal. Of course, if it comes from wind turbines, then CO2 production will be almost nil.
This source tells us how much using 1 kWh of energy contributes to production of CO2 from coal (this is the latest data I can get, but it shouldn't drift too much):
1.341 pounds of CO2 per kilowatthour generated, also showed a slight change
from 1.350 pounds CO2 per kilowatthour
in 1998 (Table 1).
That means producing 25 Wh of electricity produces 0.034 lb of CO2 (about 15.4 grams.)
An average tree absorbs from 22 pounds (1 tonne / 100 years) to 48 pounds of CO2 per year (depending on who you ask - these are quite pro-tree planting, so I'd like to see if there is a source of raw data); there does not seem to be any definite figure given.
That means one tree could support from 647 to 1,411 hand drying sessions.
But you can get far more paper from the average tree. This paper suggests that from just one cord of wood (8ft x 4ft x 4ft), smaller than a whole tree, you could produce 942,000 sheets of paper (A5 size, normal density.) Even if you used 10 sheets per session, it's still 66x better (in the best case for hand dryers!) And, you can recycle the paper used.
So no, hand dryers are not better.