What institutions are left over from the Holy Roman Empire?
The title pretty much covers it.
'Cause bureaucracy goes on forever.
Well, the Roman Catholic church would probably count, though it predates it.
Predates–does not count.
The German post office traces it’s heritage to the HRE. The Thurn und Taxis family got a Royal License way back.
The concept of a Senate?
That’s the Roman Empire. The Holy Roman one had an emperor.
“Watching your thread with great interest, seeking forwarding address since all these goddamn falcons are starting to pile up.”
Yours in Christ,
Fr. Matthew Festing,
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No, that’s the Roman Republic. Both the Roman Empire and the Holy Roman Empires had emperors (and the former also had a senate).
<SLAPS Slithy Tove with a Wet Maltese Trout>:D
Which then would be predated to the Roman Republic.
But to jump back on subject. I believe Official Permanent Embassies came to during the time of the HRE:
What society was the first to build and maintain acqueducts? The Greeks?
Bablyonians. (sp?)
The Fuggerei in Augsburg was founded in 1516 and is still operated following the original charters. Even the rent to stay there is still the same as it was in the 16th century, which is approximately 88 Euro cents in modern currency.
The Principality of Liechtenstein was established by the Emperor Charles VI in 1719, and is still a going concern.
Francis II, the last Holy Roman Emperor, retired the title and the empire to become Francis I of Austria. So I’d point to Austria as a surviving institution, and the Lippinzaner Stallions.
I really don’t know very much about Austria.
There used to be more of it, you know. 
Austria existed before Franz II became Emperor of it. The institution he created was not the nation of Austria but the office of Emperor of Austria, and that institution does not survive; it ended in 1918.
On a wider note, there would be many, many institutions - schools, universities, hospitals, subordinate government agencies, etc - founded in Imperial territories which still surivive, function and thrive. But they would mostly have been founded or chartered by the local Elector, rather than directly by the Emperor. Do they count?
Public schools
If you mean specific public schools established under the Empire which still surivive today, most of these would have been established at the local or national level, not by the Empire, which generally concerned itself with loftier matters.
If you mean the concept of a public school, i.e. an educational institution provided at communal expense which any child could attend, these long predate the Holy Roman Empire.
Does the House of Hapsburg count? They didn’t start as the emperors, but were mostly contemporaneous.