When Hogan describes how sophisticated their equipment is that is used to fool the Germans, he states that their sound recordings are "stereophonic." Stereo recordings and playback equipment would not become available to the public until the mid-1950s, well after the end of World War II.
Goldilocks on a submerged sub receives a radio call from Papa Bear Hogan. It wasn't until 1943 when US subs were able to transmit & receive radio calls without first surfacing to deploy the antenna.
When Klink lights Hogan's cigar, Hogan mouths the words "Thank you", but the word "danke" has been overdubbed.
UPDATE: When Klink finishes lighting Hogan's cigar, Hogan takes the cigar out of his mouth and licks his lips before saying, "Danke." The confusion is with Hogan licking his lips - this could appear as if Hogan is saying, "Thank" - but there is no vocalization. After licking his lips, Hogan definitely says, "Danke."
UPDATE: When Klink finishes lighting Hogan's cigar, Hogan takes the cigar out of his mouth and licks his lips before saying, "Danke." The confusion is with Hogan licking his lips - this could appear as if Hogan is saying, "Thank" - but there is no vocalization. After licking his lips, Hogan definitely says, "Danke."
When Hogan steals the cigars out of Klink's cigar box, he slips the long wire out of the hinge (it's a piano-style hinge). When he replaces the wire, it appears that he easily slides it right back into place. Given the style of hinge, it would have been nearly impossible to align the two parts of the hinge well enough to be able to slide the wire back in so easily. It also appears that his hands are not close enough to the box to actually be inserting the wire, so it is obvious that he fakes replacing the wire.
During the opening credits, an actor portraying one of the prison guards accidentally knocks over a table as he's pounding on the barracks door. The "prisoners" have to jump over and around the table as they are exiting the barracks.
Col Klink is wearing a wedding band. In later episodes, the viewer can see the "lighter" skin on his ring finger where a ring had been.
When Wagner tells Col. Burkhalter about the machine shop and mentions the souvenir lighters being manufactured there, he does not say that he has one of the lighters before Col. Burkhalter takes the cigar from Hogan - supposedly with the intent of having Wagner light it.