If the long snapper is the far left or far right man on the line, they are technically eligible receivers, so they do have opportunities to score as you can see in this example.
Also from a college punter, which I find humorous.
Also from Vanderbilt, a good example of a place kicker scoring a TD on a fake - https://www.youtube.com/watch?v=bkgmaUBNGjE
To the point where it’s a “position” that’s virtually unheard of in the NFL. 99+% of the time, the player who kicks off for an NFL team is either the placekicker or the punter. NFL rosters are too limited in size (53 man roster, only 46 can be active for a game) for most teams to even consider a player who only kicks off – that’s even more true since the kickoff spot was moved back to the 35 yard line, and most kickers can get the ball into (or through) the end zone regularly.
I can think of a few instances, over the past few decades, in which NFL teams carried a pair of kickers on the active roster. This has nearly always been when they have an older kicker, who’s still very accurate, but doesn’t have the distance any more for long field goals (or kickoffs), and a younger kicker, who has a stronger leg, but doesn’t have the same accuracy. In those cases, the younger kicker would handle kickoff duty, as well as longer field goal attempts. Usually, it seems like they’re grooming the younger kicker to take over the kicking job entirely once the older kicker retires.
In college football, with much larger rosters, teams will have multiple kickers, and it’s much more feasible to have a second kicker (stronger leg, but not with the same accuracy) who only kicks off.
As a Nashvillian and Vanderbilt alum, I’m sad that I didn’t get to post the Vandy Marinangel/Spear TDs, so I’ll just mention that I got to see a kickoff specialist in the Titans-Bulls game yesterday. Dan Carpenter kicks field goals and Colton Schmidt punts, but Billy Cundiff is handling kickoffs. (Apparenntly, Jordan Gay was the earlier kickoff specialist, but he was waived for Cundiff.)
Interesting; looking at Carpenter’s stats on the Bills’ web site, it looks like this is the second season for the team in which they have someone else kicking off. He’s been a very accurate field goal kicker, which is probably why they’re willing to keep him while burning another roster spot for a kickoff specialist.
Warren Sapp caught a TD pass. That was pretty unusual. 
Center the only one I could find was Jeff Saturday td