Not enough where you'd recover the $$$ invested within even 10 years. Lets say you use 20% less energy (very optimistic) during that 8 hour period closing off that room and assume you furanec still has enough airflow and doesn't trip and high temperature. Over a 24 hour period that's only maybe 7%. SO if you have lets say a properly sized 60k BTU furnace, you might use maybe $600 in gas over a season. That's $42/year. Is it worth all that effort to save $30-40 per year?
If you want to save $40, try eating a little less food each day, lose 15-20lbs and you're metabolic rate will drop and you will require 5% less food. Sicne your food bill is probably over $2000/year, that's $100... and you be healthier and save on medical costs.
THere's you go, I just saved you $100. I of course am joking and making the possibly unfair assumption that you can lose 15-20lbs. But just trying to make a point.
Also, in the summer, you won't save much. The ineffciency from shorter runs times due to lower load will partially offset the reduce demand. Plus the cooling load at nighttime after 12PM in most climates is minimal anyway.