It depends on the situation. In everyday "normal" life, I agree with this sentiment. However, there are times when it isn't good to "fake it." If you are going through a genuine tramatic event, you do need to have outlets for your "real" emotions, like friends, journal, support groups, etc. In day to day normal situation when people don't really care how you are feeling, faking it is fine.

Fake it till you make it has been one of my motto's, or course not told to everyone, just a mantra, otherwise people will begin to think I fake everything. When I quit drinking and smoking everything, I told myself I was just a person who did not drink or smoke, I also went to 12 step program, but the fake it till you make it threory was a big part of my recovery. That was years ago and now when I get scared or anxious etc., I still use the fact it till you make it , just to get over the fear and unpleasant feeling when change is too overwhelming. It calms me and the visualization of "everything is OK" really does help me MAKE IT.

I think there are levels. Doing your best to 'put on a brave face' to jumpstart yourself into doing positive things is a big mood booster. But, if it becomes a substitute for recognizing who you really are, or how you really feel, it can be quite crippling.
On a very superficial level, I think it is a good idea. Going to the point of suppressing real feelings is not.

No, because after a while people will begin to feel that you're just smiling to keep them from worrying.

When I was so sick for a whole year it was hard to smile, but I discovered. That if I told people I was feeling great , I began feeling that way and even if it was a fake smile, people started smiling back and that made me feel good inside and then it became a real smile.