So, yes, I believe that when a reliever gives up a grand slam to lose a game 4-3 and takes that long walk off the field, some will take a bit of that to the mound with them next time. Others won't. Most won't, really, since failure is a routine part of baseball. But to say that no pitchers are overly affected by failing in the closing role is, in my opinion, failing to consider what we know about sports psychology and baseball player mentality.
Now, the takeaway from that is not "we must have a proven closer." Not at all. My thought would be, if the guy who is Plan A hasn't closed games before, consider the history and make sure there's a solid Plan B.