Many thanks to the Third Man website for clarifying many obscure points.
Holly Martins (Joseph Cotten) goes to his friend Harry Lime's (Orson Welles) apartment – Josefsplatz 5, Vienna, Austria. The building is actually Palais Pallavicini. It’s a baroque palace built and owned by the Fries family in the late 18th century but later acquired by Alfons Pallavicini in 1842.
Martins attends Lime's funeral where he meets Major Calloway (Trevor Howard) – Zentralfriedhof, Simmeringer Hauptstraße 234, Vienna, Austria.
Martins meeting with Kurtz (Ernst Deutsch) at 'Café Mozart' – Neuer Markt 8A, Vienna, Austria. There's indeed a café there now, though the real Café Mozart is located at Albertinaplatz 2. The building on the left is the Kaisegruft (Imperial Crypt), the principal place of entombment for members of the House of Habsburg.
Kurtz shows Martins where they laid down Harry Lime after his accident – Josefsplatz, Vienna, Austria. They're standing near the statue of the Emperor Josef II (incidentally, it's him who Jeffrey Jones played in Amadeus.)
Anna's (Alida Valli) apartment – Am Hof 6A, Vienna, Austria. The old building is no longer there but the rest of the square looks pretty much the same.
Martins comes to see Dr. Winkel (Erich Ponto) – Sankt-Ulrichs-Platz 2, Vienna, Austria. It’s in the courtyard but it's hard to get in there because the front door is locked. It’s actually an entirely different building from the one Winkel later comes out with his bike.
The Casanova Club – Dorotheergasse 6-8, Vienna, Austria. The club is still there though its name is no longer on the marquee.
Dr. Winkel comes out with his bike – Börsegasse 5, Vienna, Austria
Lime’s associates meeting on a bridge – Reichsbrücke, Vienna, Austria. The original suspension bridge collapsed in 1976 and a new bridge was built in its place.
An angry crowd chasing Martins and Anna – Ruprechtsplatz, Vienna, Austria. The church in the background is Ruprechtskirche.
Martins and Anna hiding in a movie theater – Porzellangasse 19, Vienna, Austria. The former movie theater is now a theater (Schauspielhaus).
The hotel where Martins stays – Hotel Sacher, Philharmonikerstraße 4, Vienna, Austria.
A cab drives Martins to a lecture – Salesianerkirche, Rennweg 10, Vienna, Austria.
Martins running from thugs towards a church – Maria am Gestade, Salvatorgasse 12, Vienna, Austria
Martins on his way to meet with Lime – Schreyvogelgasse, Vienna, Austria
Lime lurking in the doorway – Schreyvogelgasse 8, Vienna, Austria
Martins showing Calloway where Lime vanished – Schulhof, Vienna, Austria. The basin with a cherub was apparently a prop.
The kiosk that masks the entrance to the sewers – Am Hof, Vienna, Austria. The kiosk itself was a prop.
Martins comes to see Kurtz – Morzinplatz 3, Vienna, Austria. The square has been substantially rebuilt but the building still stands. The façade has been stripped of its baroque ornaments though.
The Ferris wheel – Wiener Riesenrad, Riesenradplatz 1, Prater Park, Vienna, Austria
Anna is escorted to the railway station but then refuses to leave – Franz-Josefs-Bahnhof, Julius-Tandler-Platz, Vienna, Austria. The current terminal was built in 1978.
Martins waiting for Lime in a café – Café Marc Aurel (now closed), Hoher Markt 5, Vienna, Austria. It's a stationery store now.
The police setting a trap for Lime – Hoher Markt, Vienna, Austria.
A giant shadow appears – Reitschulgasse at Habsburgergasse, Vienna, Austria. This place is not anywhere near the Hoher Markt where the police are waiting. The shadow is projected on the Stallburg. Unfortunately, the wall was covered, and it was impossible to get a good shot.
The police are lurking among the statues – Vermählungsbrunnen (Wedding Fountain), Hoher Markt, Vienna, Austria. The fountain was commissioned by Emperor Leopold I who wanted to celebrate the return of his son Joseph from the siege of Landau. The sculptures depict the marriage between Joseph and Mary by the High Priest under a bronze baldachin, supported by four large Corinthian columns.
The entrance to the sewers – Friedrichstraße at Girardipark, Vienna, Austria. There's indeed an entrance to the sewers which was closed at the time this picture was taken.
The sewers – it was partially filmed on a studio set, partially on the covered portion of the River Wien. The actual sewer appears briefly when Lime escapes across "the waterfall."
The MPs truck arrives to another entrance to the sewers – Alserbachstraße 40, Vienna, Austria. The facades of the buildings have changed.
Lime trying to get out of the sewer through the grating – Minoretenplatz, Vienna, Austria. The grating still exists but it was moved farther down the square (it's visible in my picture.)
Anna walking away after Lime's funeral – Zentralfriedhof, Simmeringer Hauptstraße 234, Vienna, Austria. The white obelisk at the end of the road is the monument to Soviet soldiers. First, Anna is walking away from the monument, and then in the final shot she is inexplicably walking towards it - in the opposite direction.