Dangers and threats, failures and mistakes, increased pressure and ticking clocks, losing an advantage

A celebrity being mobbed by fans in public
A TV personality being recognized in compromising video footage
Committing a crime and being identified by a witness or an online sleuth
Someone noting the character's license plate as they flee a car accident
Getting pulled over by police and being seen by a relative or employer who is driving by
The character lying about their whereabouts, then being seen in the background of a news report 
An attacker being identified by their victim 
An undercover officer being recognized by a real-life connection 
A character in witness protection running into an old acquaintance on vacation
The character being seen by a friend while trying to blend into a crowd and disappear

Fandom being new to the character and so they don't know how to navigate it 
The character's children being dragged into the limelight
Having to engage with fans enough to satisfy them
The character needing to make an excuse for their whereabouts
The character having to leave the area before others also recognize them
Having to ask (or beg) someone to not mention running into them 
Needing to cut a visit or outing short because a cover is blown
Trying to pretend to be someone else: Oh, I get mistaken for Henry Cavill all the time.
The character denying that they are who they are
Fielding awkward questions from family members
Needing to avoid cameras and people with phones
Being forced to keep a low profile
Having to silence a witness without getting caught
The character needing to find out how much the person who recognized them knows