In the future eternal world of Olam Haba (i.e. the final destination following the resurrection of the dead as explained here) according to the opinions that the body and soul will be reunited, will there still exist plants and animals?
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The Leshem writes that any object, including inanimate objects, which were part of bringing about a kiddush Hashem or a revelation of Hashem's presence in the world, will exist in the eternal Olam Haba.
According to the Rambam, Olam Haba contains nothing physical. See Hilchos Teshuva 8:2:
העולם הבא--אין בו גוף וגווייה, אלא נפשות הצדיקים בלבד, בלא גוף כמלאכי השרת. הואיל ואין בו גווייות, אין בו לא אכילה ולא שתייה ולא דבר מכל הדברים שגופות בני האדם צריכין להן בעולם הזה.
In the World to Come there is no body, only the souls of the righteous alone, without bodies, like the angels. Therefore, there is no eating or drinking, or anything that human bodies need in this world.
Similarly the Rambam explains in Yesodei HaTorah 4:1-9 that everything except the soul is made up of component parts and will therefore break down, whereas the soul is [directly] from Hashem and will therefore last forever (the implication being that nothing else will last forever).