It has been a while since I looked into my econometric books but I thought it could go either way depending upon the marginal impact of the added variable. Addition of independent variables will cost degrees of freedom but will also have added explanatory power and thus, increasing the ESS and decreasing the RSS.

Try running a regression with one irrelevant independent variable and adding the relevant variables after that. What does the model say for F-stat? My take would be, F-stat will increase. I might be wrong as well.

By the way, here is a link to the material on introductory econometrics by a LSE Professor if you want to look:

http://econ.lse.ac.uk/courses/ec220/G/ieppt/series2/#chap1