Will A in comments notes the lack of research about how much AV would affect the number of 'safe seats', and has a go at producing an answer.
It's an interesting question, so the swingometer now includes a table of majorities in its results. Incidentally, because Uniform National Swing can give parties a negative number of votes in certain constituencies, which I am only partly able to compensate for when calculating transfers of votes, the majority calculations under UNS are probably not as good as those under Proportional Swing.
The results are fairly inconclusive - depending on what transfer patterns are chosen, whether the seats as a whole get safer or not is really difficult to tell. Certainly, while in general the majorities are reduced more than they're increased, this is mainly through changes from one large majority to another slightly less large majority. The number of seats in the top few rows of the table doesn't change much.
So, in that sense, no - AV won't do much towards (or against) eliminating safe seats. It will change which ones are which, though that doesn't make a lot of difference.
The question is, is the majority of a seat the correct measure of how safe it is? It sounds an odd question - that's the way that the safety of seats is normally measured - but one of the major problems of trying to estimate general election results based on extrapolating national polling to individual constituencies is that you have to make the assumption that there are no local effects whatsoever.
This, of course, isn't true, but "hoping the resulting errors cancel out" is actually surprisingly effective in UK general elections.
For working out how "safe" a seat is, rather than who will win, it's probably one extrapolation too far. One thing we really don't know yet1 is if the transfer paths of (for example) Lib Dem voters in Con-Lab marginals (where their transfer decisions could make a big difference) are the same as those of voters in Lab-Lib marginals (where their transfer decisions are probably irrelevant unless the Lib Dem candidate has an electoral disaster)
If, for instance, there was a big anti-incumbent effect in second preferences, it probably wouldn't be very noticeable in terms of discrepancies from national polling - across the country it would mostly cancel out - but it could make a big difference to the sizes of majorities in seats.
Consider, for instance, the Tatton constituency in 1997. Neil Hamilton had held it with well over 50% of the vote for several elections, and under normal circumstances would have held on in 1997 too with a decent though reduced majority - despite the Conservatives' national unpopularity. He was defending a majority of 15,000. Indeed, in 2001, when the Conservatives popularity was basically unchanged from 1997, George Osborne was able to take the seat back with a comfortable majority when Martin Bell declined to re-stand.
Of course, that's not what happened - Hamilton, in addition to his party's unpopularity, was part of a major political scandal. Martin Bell stood as an "anti-corruption" Independent, and the other major parties decided to go for the FPTP equivalent of Alternative Vote by not contesting the seat.
Hamilton's 15,000 safe majority was replaced by a 11,000 majority for Bell.
Under Alternative Vote, it's fairly clear that Hamilton would have lost even if Labour and the Lib Dems had stood candidates - whether he'd have lost to Bell or to one of their candidates, who knows - but this wouldn't have shown up on any national calculations of seat safety.
(Martin Bell went on to contest Brentwood and Ongar in 2001, but the other parties didn't stand aside, and he lost narrowly to Eric Pickles MP - a loss that was just another expected result at the time, but in retrospect has had major repercussions)
To conclude: under normal circumstances, AV will not make any significant difference to the number of safe seats. Under local circumstances where the incumbent has significant personal unpopularity2, though, pretty much any MP could be vulnerable, even those who would normally get 50% on first preferences alone.
How many FPTP-safe seats actually change hands under "local circumstances" under AV, is of course by definition virtually impossible to predict. It probably won't be many at each election; on the other hand, it will be very worthwhile for the voters in those constituencies if it does happen.
Footnote
1 Unlike many of the unknowns, this isn't one that could be cleared up by comparing polling to an AV election, since the election results won't include transfer information for votes that didn't transfer, or preferences that got skipped due to earlier eliminations. Trying to work out if second preference polling was even accurate will be a substantial challenge.
2 Or a major local issue that an independent candidate could exploit, as Dr Richard Taylor did in Wyre Forest
