No, you can't It's always: "Long to do something".

Yes, the 2nd sentence is awkward. "longing to visit" is the correct phrase to use. You can only "long for" an object. That place has been longing for me to visit. (The meaning of the sentence is changed here, but this is how "longing for" could be used.)

It looks like they say "long to do something" and "long for something(noun)", I long for your love... that's very old fashioned, man.