Firstly, finding out the flavor of the beans would require divination. Professor Dumbledore was a very powerful wizard, but I believe he wasn't very good in divination. He admits himself in Half-Blood Prince chapter 20:
[…] ‘Divination is turning out to be much more trouble than I could have foreseen, never having studied the subject myself. […]‘
And also in Prisoner of Azkaban chapter 22.
‘[…] The consequences of our actions are always so complicated, so diverse, that predicting the future is a very difficult business indeed … Professor Trelawney, bless her, is living proof of that. […]’
Professor Dumbledore used his great people skills, mind-altering magic, and network of spies to find out a lot of information about the past history and future plans of the Dark Lord, but it doesn't seem like he's ever used divination for that.
(Update: on second thought, it's hard to be sure, because Hermione, who has quit Divination lessons early, has still cast at least one Divination spell in Hallows.)
Professor Dumbledore may have had friends in the Order or Hogwarts who could have helped him in divination magic, and it would have been even in character for him to ask them a favor for finding out the flavor of some every-flavored beans, but he chose not to do that for some reason.
Secondly some magic systems have ways to explicitly block divination. These certainly exist already in the Harry Potter universe too, because the Order of the Phoenix could successfully protect its secrets from even the Dark Lord, who is a very powerful wizard and also had powerful allies. The Fidelius (secret keeper) charm was one method used for this, but I believe there were others, including possibly house elf magic.
Case in point, Hermione has cast some divination blocking spells when they were on the run. This is revealed clearly in Deathly Hallows chapter 19.
‘One thing I would like to know, though,’ she [Hermione] said, fixing her eyes on a spot a foot over Ron's head. ‘How exactly did you find us tonight? That's important. Once we know, we'll be able to make sure we're not visited by anyone else we don't want to see.’
‘Yeah, well, that would've been me,’ said Ron. ‘Your protective spells work, anyway, because I couldn't see you and I couldn't hear you. I was sure you were around, though, so in the end I got in my sleeping bag and waited for one of you to appear. I thought you'd have to show yourselves when you packed up the tent.’
Thirdly, let me point to a very similar example, from later than Philosopher's Stone, of divination failing in such a frivolous way. In The Order of the Stick strip #413, Haley points out in panel 13 that the royal court of Azure City could not buy booster packs that contain the right pre-painted minifigures, despite that they have a very good sage. This is probably common in the Order of the Stick universe, because there are shops making a business from selling spells in randomized booster packs to wizards, many of whom are likely good diviners.
Update: I added the quote from Prince chapter 20, removed a less relevant quote, and reorganized the text around them a little. Thanks to TenthJustice for pointing out this quote in Why did Dumbledore contemplate discontinuing Divination at Hogwarts? and to Slytherincess for locating the quote.