Here's the solution I've ended up with, based on your suggestion:

CREATE FUNCTION skip_update() RETURNS trigger AS $$
    BEGIN
        RETURN NULL;
    END;
$$ LANGUAGE plpgsql;
CREATE FUNCTION fail_update() RETURNS trigger AS $$
    BEGIN
        RAISE EXCEPTION 'can''t modify row values for existing primary
+ key';
    END;
$$ LANGUAGE plpgsql;
CREATE TABLE Dummy_Test (
    Timestamp TIMESTAMP WITH TIME ZONE PRIMARY KEY,
    Foo TEXT,
    Bar TEXT
);
CREATE TRIGGER Dummy_Test_dupe BEFORE UPDATE ON Dummy_Test FOR EACH RO
+W
    WHEN ( OLD.Timestamp IS NOT DISTINCT FROM NEW.Timestamp AND OLD.Fo
+o IS NOT DISTINCT FROM NEW.Foo AND OLD.Bar IS NOT DISTINCT FROM NEW.B
+ar )
    EXECUTE FUNCTION skip_update();
CREATE TRIGGER Dummy_Test_modify BEFORE UPDATE ON Dummy_Test FOR EACH 
+ROW
    WHEN ( OLD.Foo IS DISTINCT FROM NEW.Foo OR OLD.Bar IS DISTINCT FRO
+M NEW.Bar )
    EXECUTE FUNCTION fail_update();

INSERT INTO Dummy_Test (Timestamp,Foo,Bar) VALUES ('2020-01-02 12:34',
+'Hello','World') ON CONFLICT ON CONSTRAINT Dummy_Test_pkey DO UPDATE 
+SET Foo=EXCLUDED.Foo, Bar=EXCLUDED.Bar;
INSERT INTO Dummy_Test (Timestamp,Foo,Bar) VALUES ('2020-01-02 12:34',
+'Hello','World') ON CONFLICT ON CONSTRAINT Dummy_Test_pkey DO UPDATE 
+SET Foo=EXCLUDED.Foo, Bar=EXCLUDED.Bar;
INSERT INTO Dummy_Test (Timestamp,Foo,Bar) VALUES ('2020-01-02 12:34',
+'Hello','abcde') ON CONFLICT ON CONSTRAINT Dummy_Test_pkey DO UPDATE 
+SET Foo=EXCLUDED.Foo, Bar=EXCLUDED.Bar;
[download]

The last statement will fail, which is exactly what I wanted. The function FUNCTION skip_update() and TRIGGER Dummy_Test_dupe really only prevent the UPDATE from happening, they can be omitted - I haven't yet tested whether it's more performant to have the TRIGGER or the UPDATE fire. In any case, thank you very much!