It is confusing to teach beo as a new adjective when it seems to be proposed first here as a noun with the copula (same case with fíor), even if a correct English translation turns out to be the same in both cases with the copula or bí (or maybe I am missing something again, which is entirely possible - I'll check back notes).
edit: ok found my mistake, beo is used after turtle (and thus an adjective), and not after the copula or the é (which would make it a noun).
This is one of those cases where the dictionary entry might be a bit overly specific. It gives two very idiomatic examples, na mbeo agus na marbh and beo ná ceo, but I'm not sure that beo would normally be used to mean "a living being" as in your sentence (where "living" is still an adjective).
I'm not saying that Is beo é an turtar is definitely wrong, just that I'm not sure that the entry in focloir.ie really justifies it's use that way, and I'd look for some other sources before using it that way.
"alive" is an adjective. In "The turtle is alive" you have a verb, a noun and an adjective - tá an turtar beo.
In "it is a live/living turtle", you have a pronoun ("it") and a noun ("turtle"), so you use the copula rather than tá - Is turtar beo é.
The adjective beo is not really relevant to that grammatical decision - you can just leave the adjective out and get "It is a turtle" is Is turtar é.