Aaron, your guess is correct: the produce remains kosher whether it was shared with the poor or not.
The only portions of the produce that have restrictions on its edibility are:
- terumah, which must be eaten by a Kohen while ritually pure
- terumat maaser, which is the terumah given by the Levi.
- Ma'aser sheni, should be kept ritually pure (tahor) and eaten in Jerusalem.
Hence, the way Israeli produce is tithed today is as follows:
Okay here's 100 lbs of wheat. I take this one kernel and declare it terumah, I can't eat it as I'm not a kohen, but a kohen can't eat it as we're likely all tamei, so I'll just let it decompose. Now I designate the northern 10 lb of this wheat as maaser, which should be given to the Levi. Okay the Levi should give a tenth of that to the kohen, so the north-eastern 1 lb of this wheat (remove it) is hereby terumat maaser, again I just have to let it decompose. The other 9 lbs on the north are ordinary maaser, they're kosher for me to eat; if anyone can prove they're a Levi, come and get them. What, nobody here? Oh well. I'll just eat them myself then.
The southern 10 lbs (well slightly less) are ma'aser sheni, I should take that wheat up to Jerusalem and eat it there, but it's tamei already, so I hereby transfer its sanctity onto this handy coin (which, as I can't use to buy non-tamei food in Jerusalem, I have to throw out). Okay, now I can eat the southern 10 lbs.
The shares given to the Levi and poor are all about monetary ownership, not ritual status of the food. If gifts to the poor affected ritual status, we'd have to get into a complicated question of who's called sufficiently "poor" as to have a ritual effect. Yes the Talmud defines poverty with regards to who's entitled/allowed to receive these forms of charity, but how much money you have in the bank, to the best of my knowledge, does not affect ritual status in halacha. It's not something directly tied to who we are, no matter what advertisers would like you think otherwise.
(Yes okay nitpickers, what if someone makes an oath or marries a woman "on the condition that I'm rich" ... yes halachic actions can be conditioned on anything.)