Having visited 190 at the time of writing, I often am asked if I plan on visiting all of them.
The answer is “no” and I’m not even sure it is possible. As of 2013 there are 962 UNESCO World Heritage sites and about 1-2 dozen are added every year. It would be difficult to keep up with the new entries on the list, let alone making a dent in the 900+ already listed.
The most I know of that any one person has visited is Bill Altaffer who has been traveling for over 60 years and claims to have visited visited 808. Iain Jackson of Scottland has visited 691 world heritage sites. He has been visiting them since the 1980’s and claims that his unvisited list is now larger than it was in the 80’s just because they keep making so many.
Many World Heritage Sites are easy to visit (Statue of Liberty, Paris, Rome, Westminster Abbey). However, some are extremely difficult to visit and require a determined effort to get there. There are several sites I’ve attempted to reach but failed (I’m looking at you Archaeological Sites of Bat, Al-Khutm and Al-Ayn in Oman).
It would probably take a lifetime of travel and millions of dollars to seriously attempt to visit every single one on Earth.
Even if you had the time and money, however, you’d still probably fail in your quest because some UNESCO sites are almost impossible to reach.