From an older answer
In one of the least exciting answers I can give you: Nothing happened to them.
Presumably many changed their name although this is at the time not possible to confirm since German Personenstand files (meaning files of the agency that handles name changes and such) are restricted in access for privacy protection reasons usually for either a 110 years after their birth, 30 years after their death or 80 years after their name change and thus – unless you can prove valid scientific inquiry such as family research or an academic research project – it's not possible yet to verify just how many changed their names.
German law in general allows for a change in last name only in a very specific set of circumstances although one of them that is when a you have a name that is the same as a criminal and that is so rarely used that extensive reporting on the crimes of said criminal is likely to cause negative connotations and consequences for the person carrying the same name. Which in the case of Hitler was certainly warranted.
And finally, there weren't many people named "Hitler" to being with. Adolf Hitler's father, Alois, had been born an illegitimate child and originally carried the name Schickelgruber. When he was ten, his mother married a man called "Hiedler" who didn't outright adopt him but functioned as his father. Later in life, when Alois was married and already a customs official in Austria, the man who had functioned as his father, Hiedler, made it a condition of his testament that Alois change his name and so, in 1876 (when laws concerning name changes were less strict as they are today), Alois changed his name from Schickelgruber to a variant spelling of Hiedler, Hitler.
While no concrete numbers exist to my knowledge, a quick research in the Vienna phone book of the 1930s reveals that in Vienna, Austria's largest city, there were 7 people named "Hiedler", one named "Hiettler", one named "Hitler" (Paula, Adollf's sister), and none spelled "Hittler", which also sometimes occurs. And while the name "Hitler" wasn't exclusive to the family of Adolf, it is very, very uncommon as a variant of Hiedler. It's impossible to know how many people are still named that way but the contemporary phone books for Austria and Germany list only one person and a blog post I couldn't further substantiate (thus to be taken with a grain of salt), said that there are about 2800 people with all variants of the name, including "Hiedler", "Hiettler", "Hitler" and "Hittler".
In essence, the name was not very common to begin with and after WWII we can presume with a reasonable degree of certainty that a lot of people with the name did change it thus making it even less common. Nothing happened to the all the people named Hitler out there, except that for understandable reasons, they didn't want to carry the same name.
At the last months of the war or partly after the war most of the German people did not want to be associated with the NSDAP (Nazi political party) or any High ranking Nazi members (for obvious reasons). Especially after the Allies issued the denazification of Germany and the Nurnberg Trials took place many people, who just then realized the scale of crimes on humanity comitted by its nation. (important to note is that most of the people could have known or did know about concentration camps, but decided to actively look away when Jews got deported or even took part in the process).
This led to many people involved in the process trying to hide their ties to the Nazi party by burning their NSDAP membership books. (also because they feared persecution by the allies) People who shared a surname with high ranking Nazi party members also tried not to be associated with the crimes comitted and the complete destruction of their home country. Mostly it did not matter if the people were relatives to the Nazi members or if they just chared the surname coincidentally.
As part of my history class we did discuss the matter of Reichspropagandaminister Joseph Goebels. This surname is quite interesting because it was a very common surname before WWII. In order to not get associated with him many people changed their surnames to:
The name Goebel and Gebel were very common in Germany and are to this day. (Just to avoid missunderstanding there were people called Goebel or Gebel before WWII).
As you can imagine changing your name after WWII was easier then today (without internet, many peoply loosing their passports or burning it).
We discussed that all of the surnames by the Highest ranked Nazi members are almost not to be found in Germany for this reason. Unfortunately i can not tell you anything about the surname Hitler in particular, however i hope i was able to give you an basic understanding that people tried to change their names. Thus resulting in Hitler or Goebels not beeing German surnames anymore.