Heating in winter is ok but when you can't turn it off in summer it's al little bit too much.
Any idea where to look for the problem?

cable could be too tight so its holding the flappers open, check the tension on those cables that run to each flapper box , i usually remove the tubes and put caps on both ends so no air is flowing from the HE to the tubes up front .

Phil

Oddly a normal spray paint can cover fits perfectly on the heater tube end sticking out of the longitudinal, and it keeps out rodents.

It is also possible someone shoved the flappers on more than necessary and the flap is conflicting with the heater tube. That happened to me; I loosed the flapper valve and pulled it back about 1/4" and the flapper opened and closed as it should.

Disconnect the cables to each side at the flapper lever and operate the flapper by hand from under the car to see if it is opening and closing completely.

Was someone making flapper box rebuild kits?

Control wire too tight? Set the operating length with the handle down (fully closed position). The gaskets might be spent or missing.

Dansk makes them. Around$100.