The easiest one is the hype that is surrounding him. He was selected with the 6th overall pick in the 2009 draft, a draft that saw Stephen Strasburg go number one overall to the Nationals. The Giants selected him right after the Orioles took RHP Matthew Hobgood, who is now a 22 year old pitching in high A ball. Wheeler was taken over LHP Mike Minor, who went with the 7th pick to Atlanta, as well as RHPs Mike Leake (8th to Cincinnati), Jacob Turner (9th to Detroit) and Drew Storen (10th to Washington). While some pitchers taken that high never make it to the majors, Wheeler has been expected to be as good as advertised.
When the Mets were expecting to bring up Harvey and Wheeler up to the big leagues, many scouts said Wheeler had the higher end stuff. It was expected that Wheeler would become the better pitcher. Much talk was about Wheeler even after Harvey was called up last season. Harvey (8-2, 2.11, 164 Ks in 145 IP) has pitched at an elite level in just his first MLB season. His starting of the 2013 All Star Game was an indication of how quickly he is now being mentioned among the top starting pitchers in all of baseball. Of course, Harvey has to continue to pitch at this level. But Wheeler now has to live up to his own potential and shouldn't have to think about how Harvey is doing. He obviously has a ways to go if he wants to get to Matt Harvey's level.
Next is something Zack Wheeler should never think about. If he does, it may put up a caution flag about his mental toughness. Sandy Alderson has put a lot of eggs in the Wheeler basket. He traded for Wheeler and outside of the RA Dickey trade, has done little else in bringing in impact players who can be considered his doing. The Mets are continuing to run out a large group of Omar Minaya's players. Wheeler's success or failure will have a lot to say about whether Alderson has done a good job or not. Wheeler as an ace, or a 1-A with Harvey, will make Alderson look like a genius. If he fails, more people will start to question what Alderson has done since he took over as GM of the Mets.
None of these burdens should be in Zack Wheeler's mind at all. Nor will they be. But the media will still make him out to be the next Seaver, the next Harvey or the next Dwight Gooden. It is on Wheeler to focus on being the best version of Zack Wheeler he can be. He needs to continue to work on his command and not worry about having a couple bad games. Once he lets his ability and skills do the talking, he will be placed where he deserves to be placed. Wheeler has pitched good enough so far, but should continue to improve as time goes on. If he does, all three of the burdens will be unintentionally overcome.