From a vocabulary perspective, reading Biblical Hebrew (BH) from a Modern Israeli Hebrew (MIH) perspective is probably somewhat akin to reading Shakespeare (but see cautions below about meanings). However, there are grammatical constructs and language nuances that make it tricky, leading a Hebrew professor I studied with to characterize reading BH without any BH education as like reading old English. Once you learn these differences the problem is probably largely averted. (Note: my study of MIH has been limited to something like 60 classroom hours; my focus has largely been BH.)
In the rest of this answer I'll be quoting from Biblical Hebrew for Students of Modern Israeli Hebrew by Marc Zvi Brettler, professor of Hebrew Bible at Brandeis. This book is used by rabbinic students at Hebrew Union College and, presumably, other places. It is aimed at students who already know Modern Hebrew.
From the author's preface:
Yet BH and MIH are two different languages -- or at the very least, two substantially different dialects of the same language. MIH is certainly useful for reading the Bible, but no one can understand the Hebrew Bible knowing only MIH. There are significant differences in vocabulary, spelling, verb formation, use of verbal suffixes, and word order.
Some constructs in BH are unknown in MIH, such as the vav conversive (reversing vav), which converts a perfect verb to imperfect or vice-versa. (BH doesn't really have "past" and "future" tenses so much as perfect and imperfect aspects -- another key difference.) Without knowing about this often-used construct one would make significant errors, some but not all of which might raise contextual alarms ("that doesn't make sense").
Some constructs are different. MIH tends strongly to subject-verb-object order and (to my understanding) doesn't use the direct-object marker et. BH, on the other hand, is less consistent; it tends to put the verb first with the subject and object following, but sometimes starts with the object. Since the verb construct tells you the number and gender of the subject, subjects are sometimes dropped as redundant, which can lead to ambiguity.
BH and MIH handle participles differently, and also some verb prefixes and suffixes. The book has extensive discussions of this. Some vocabulary is also different. While MIH vocabulary is derived from BH vocabulary, knowing only MIH doesn't always get you back to the core.
BH itself isn't completely uniform either. From the introduction:
The second problem alluded to in the term biblical Hebrew was that it implies that we have a single, unified language. On the contrary, we actually have several dialects that are merged in the Hebrew Bible. These dialects may be distinguished mainly in terms of chronology, geography, and genre.
He goes on to point out that the language changed over the span of a thousand years, particularly through exile when Aramaic had a stronger influence; that there were differences between the northern and southern kingdoms after the split; and that poetry is rather different from prose.
Throughout the book he offers some translation tips, which I quote from to illustrate the types of problems that can arise. (I'm going to sometimes transliterate for ease of composing this answer.)
Do not assume that a word has the same meaning in BH and MIH. Example: Song of Songs 4:1 שַׂעְרֵךְ כְּעֵדֶר הָעִזִּים, שֶׁגָּלְשׁוּ מֵהַר גִּלְעָד. The word galash in MIH means "ski", but here it means "descend".
(You can see how, given a root for "descend" and a need for the concept of downill skiing, the creators of MH might have applied the former to the latter. This isn't a unique example by any means.)
Vowels matter. In particular, be sure to distinguish between verbs and nouns. Example: Psalm 13:6 אָשִׁירָה לַיהוָה, כִּי גָמַל עָלָי. Gamal with a qametz is the noun "camel", but gamal with a patach as here means "deal fairly" or "reward".
I understand from this book that vowel-driven nuances are way more common in BH.
He also points out issues with different classes of defective verbs, and with verbs that look the same in different binyanim (so you need to apply context).
A speaker of Modern Hebrew who is aware of the differences between it and Biblical Hebrew should be able to read BH about as well as we read Shakespeare. A speaker of Modern Hebrew who hasn't internalized those differences will make many mistakes. He will probably not realize he is making those mistakes, however; the language won't be cryptic to him, the way Old English would be to most of us. So neither of these is really the right analogy, but I don't have a better one to suggest -- a flavor of English that you think you understand and don't (unaided).