It sounds like a pretty simple (if not obscure) question. The problem is, this is pulled from a Simpsons quote, and googling anything remotely related to this question, only pulls you back to the quote.
Anyways, was there ever a time in history when this quote may have made sense? What is an autogyro, and was it ever used to deliver mail?
Prussia existed 1525 to 1932.
Siam is messier, only because Siam wasn't a country distinct from Thailand. It was simply what outsiders called it. Officially it had that name in some capacity until 1939, and briefly after but that doesn't matter to your question. We can take Siam simply as referring to Thailand when used by foreigners, at least until the end of Prussia.
So now all that matters is whether they had diplomatic relations (and the autogyro part but I don't think we'll need to get that far). I have good news for you. Formal diplomatic relations were established between the countries in 1862. Unfortunately, the bad news is there is no indication that they ever set up a regular diplomatic mission in the capital. After establishing the treaties, the embassy once again left Thailand.
So while it was the start of formal relations (and a continuance of the sort of imbalanced treaties that European powers were so keen on at the time), it wasn't long-lasting at least in terms of having a physical address in Siam. Shortly after the Prussian mission to the region, domestic politics in Europe shifted and the push to establish an overseas presence was drastically cut short. What's more, future diplomatic missions (of the stationary sort) were more focused on the German states as a whole with Bismark at the helm, occurring almost immediately after the mission to Siam.
So, sadly, autogyros aside, the letter would not have been able to have been sent and Burns would have surely known this given his age.
Anyways, was there ever a time in history when this quote may have made sense?
The only time in history the quote would have ever made sense is if Burns was speaking specifically to sending a letter to the embassy (as in the group of people), referring to them as a consulate, and speaking of a very small window of time around 1861-1862. Beyond that, it would not have had any reasonable meaning.
You can read a bit more about this if you're interested in Martin's The Prussian Expedition to the Far East (1860-1862), which is freely available online with minimal Googling.
Sorry I can't help much with the autogryos.