I've been studying paleography for about a year now, and I really don't find this to be true. This seems to be a common story, one that I've heard before and one which makes plenty of sense, but also one which seems to be false. Certainly, the use of <y> in place of <þ> for certain common abbreviations would've become most common in the early days of the printing press—not because presses couldn't have a thorn character, but because it didn't make sense to spend resources making one if it wasn't necessary. However, the substitution of <y> for <þ> in scribal abbreviations started happening in English manuscripts even before the introduction of the printing press in England.
In the most common scripts at the time (cursiva anglicana and later bastarda anglicana), <y> and <þ> were already very similar (the only difference being which of the two strokes had the descender—both letters have a closed bowl in these scripts), so similar that it was common in bookhands to write a small dot above <y> to aid the reader even when <þ> was still visually distinct. You can see this clearly in [1], written c. 1399. Use of either character to represent <th> was already becoming quite rare in the late Middle English period, and remained in use almost exclusively as a scribal shorthand; it's not uncommon to encounter manuscripts in which <þ> and <th> are mixed freely.
I can't say for certain why it is that <y> started to be substituted for <þ> in manuscripts. Obviously in the common hands of the time they're visually very similar, yet I'd imagine anyone who'd been educated as a scribe would be able to spot and write the difference. Perhaps it was a stylistic choice. But it did happen: you can see an example at [2]. This document was written c. 1445. Gutenberg's printing press is thought to have been invented at this time, but it wouldn't arrive in England until Caxton set one up c. 1476. It seems very unlikely to me that the printing press had anything to do with the initial substitution, though I'd imagine that the press was a major nail in thorn's coffin.
I definitely agree that printing was the main cause of the usage of <y> becoming commonplace, and it's likely because printing substantially lowered the cost of written material—both producing and purchasing. It enabled widespread dissemination of written documents for the first time, so any conventions printers happened to adopt then coincidentally became widespread. The arrival of the press is similarly credited with the development of a standard written form of English, though that would take some time to complete.
I spent some time poring over facsimiles of early printed English books. Caxton is credited with printing the first book in English, actually a couple years before he returned to England, while he was still in Flanders. In that book, it looks to me like there's two glyphs, one which resembles <þ> and one which resembles <y>, but they're used completely interchangeably as if they're the same character. However, pages printed on an actual press usually wound up smudging a bit when the page was peeled off of the block, so the differences I perceived could very well just be artifacts of the printing process rather than intentionally distinct glyphs.
That said, early books that were printed in England do have a noticeably distinct <þ> character, but it's use is limited to the common scribal abbreviations, which themselves are quite rare in this material. I couldn't pinpoint any precise date for <þ> -> <y>, but after around 1560, those old scribal abbreviations seem to disappear almost entirely (I only found one document after that which had any such abbreviations—from around 1680(!)—and this document unmistakably uses <y>). It's always apparent whether <y> is intended in its modern sense or as a substitute for <þ> because the abbreviations are always indicated with a superscript.
I wonder if abbreviations in certain types of work were viewed as un-ideal. They certainly seem less common (but still present) in longer works and in works which seem to be a bit more formal. In short works, especially where space was at a premium (broadsides, chapbooks, newsletters, etc.), they are more common, which is understandable. This also seems to be a continuation of a trend that started in manuscripts, where fancier and more elaborately-illuminated texts started employing abbreviations less often. But at the moment I haven't the foggiest why they appeared at all in longer works—especially when print comes into play—since their usage is rare and doesn't follow any obvious pattern—perhaps the typesetter was just running out of <t>'s!
The picture is pretty muddy, and further complicated by the timing of the press's appearance because we can't really know if the development of <þ> -> <y> would've happened anyway, or if it would've simply remained a quirk of certain scribes.
Anyway, when I woke up this morning I had no idea that this was how I was going to spend my day, and it was fun to dig through all this stuff :)
Slightly off-topic: while I was busy skimming facsimiles for abbreviations, I came across this gem, which I find both interesting and... "irrationally amusing" (I don't know how else to describe it, but if I weren't so robotic I might've been giggling like a child):
http://hdl.huntington.org/cdm/singleitem/collection/p15150co...
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