The phrase Happy Birthday has spread way beyond the English language, and is commonly seen and heard in many different countries. It is not surprising that it is often mispronounced, in particular as it contains the tricky th. A quick youtube search produced the following video (which was certainly made with the best intentions and well received, and whose message should in no way be diminished by this post),
In addition to the correct pronunciations [ˈbɜːθdeɪ] (British) and [ˈbɜːrθdeɪ] (American), listen here, the following variants can be heard in the video [the (r) takes care of the British and the American versions]:
[ˈbɜː(r)ðdeɪ], with the th pronounced as in the, listen here. As discussed previously, noticing and distinguishing the different possible pronunciations of the th is a key step toward becoming an advanced English speaker.
[ˈbɜː(r)ðeɪ], with th again pronounced as in the, but completely leaving out the d. Compare this to the correct pronunciation of birthday given above, in which the d is clearly present. Whereas letters are often dropped in compound English words, birthday is one example in which the pronunciation of birth and day is unchanged compared to the individual words.
[ˈbɜː(r)sdeɪ], with th pronounced in the German way, namely as [s]. This incorrect pronunciation can be quite confusing, see the video below: