This example occurs in Iliad 1.33. In running speech, when there are no pauses between words, I'm able to articulate this as "edeisend ho." However, I would imagine (possibly just because I'm ignorant) that part of the definition of a "word" in linguistics would be that one can choose to pause between words without making one's speech incorrect. Personally, I am not able to articulate "edeisen dho." All the same issues seem to arise with τέ as with δέ.
My guess would be that one of the following is true:
The elision only exists in running speech, so a speaker who chose to pause after ἔδεισεν would have to say "ἔδεισεν δέ ὁ γέρων." This messes up the meter, so when actually performing this line, one could introduce a principal caesura at this point (not that it's logical), but it would then be necessary to add a "pick-up" syllable after the pause.
Greeks could pronounce word-initial consonant clusters that I can't pronounce, and this is one of them.
The postpositions τέ and δέ are very strongly bound to the word before it, like "'s" in "Bob's." Writing a space in between is a spelling convention, but it really acts more like the ζε in Ἀθήναζε. You just can't pause after ἔδεισεν.
Even if we don't pause between words, the aspiration following this elision is dropped, so although we still spell the word ὁ, really this phrase is properly pronounced "edeisendogeron."
Similar to 4, but the pronunciation becomes something else, maybe θο γέρων.
Re #5, searching on the text of the Iliad shows that although it's quite common to have stuff like τ' ἀνδρῶν and τε ἀρνῶν, one never sees τ' followed by a vowel with rough breathing, which would presumably have resulted in a θ pronunciation.
Do we have any way of knowing which of these, if any, is true? Since δέ and τέ are extinct, I guess we can't tell based on modern speech. Since they're postpositions, there are no cases where they're the first word of a sentence. Searching through the text of the Iliad, I also don't find any cases where δέ or τέ is the first word on a line (which would seem to support possibility 3 above, since it was probably normal to pause at the end of lines).
As a side note, it seems that δ and τ are never word-final in Greek, except in κάδ, a synonym for κατά, and this seems to occur only in "κὰδ δέ."