One objection to God's omniscience is that He can often be seen, throughout Scripture, either asking questions (e.g. "Where are you?" in Genesis 3:9), or saying things will come to pass which, ultimately, do not (e.g. "Set your house in order, for you shall die, and not live." in 2 Kings 20:1). If God is omniscient, it is reasoned that these things should never have left His mouth. Why, if God knows all things, should He ask questions? Or why, if He knows the future, should He make apparently erroneous declarations about it?
In truth, however, there are many reasons one might conceal, downplay, or otherwise play dumb about the knowledge one has.
To illustrate this, consider the modern phenomenon of video games. In the past, games like battleship were always played against other people, who genuinely do not know the location of your ships. You tell them where to place their pins, they tell you if it landed on their ship or not (and if that "sunk" it). And vice versa. Now, however, we have the technology to play against a computer. The very same computer on which we play.
Because we are playing on the computer, the computer knows, at every given moment, where your ships are. But that wouldn't make a very fun game, now, would it? Either it would always win, or it would have to show you its board and you'd just take turns hitting each others' ships with 100% accuracy. Thus, the computer, for sake of both fairness and fun, will pretend it doesn't know where you have placed your ships.
When it comes to God, He isn't "playing dumb" for sake of fun and fairness in a game. Rather, His questions always have a purpose. Beginning in Genesis, there are actually more examples of God asking questions than just Genesis 3. In Genesis 3, God asks questions to illustrate, both for Adam and Eve, and for us here in the modern day, the true nature of sin. It is a convicting thing. Adam and Eve, while they did not give the answer they should have given, nevertheless answered in ways which are useful to us, showing the convictions they felt, but also showing the poor excuses they gave.
Later on in Genesis, God tries the same tactic with Cain, but Cain doesn't respond in the same way. He does not, as Adam and Eve, show his guilt by trying to conceal his shame, but nevertheless admitting to God his fear. Rather, Cain lied, trying to conceal his guilt entirely. "Where is Abel your brother?", God asks, to which Cain replies "I do not know. Am I my brother’s keeper?" (Genesis 4:9). But rather than turning aside to go looking elsewhere for Abel, as if "where is Abel" is a genuine question, God informs Cain that He not only knows where Cain is, but that Cain is the culprit. Thus, without hesitation, He punishes Cain severely.
The simple fact is, by asking questions, God is not revealing His ignorance. Rather, by asking these questions, He is prompting certain responses. Are we expected to believe that Jesus, when holding a Denarius, did not know Caesar's portrait (Matthew 22:20; Mark 12:16; Luke 20:24)? Or was He merely guiding the Jews to understand His point, even circumventing the trap they had laid for Him? This was not Jesus demonstrating His ignorance, but His great wisdom!
But this still doesn't seem to explain God's giving of prophecy that never comes to pass. In truth, however, it does. In similar ways, though not entirely identical, God's telling of possible futures is more of a prompt. In the case of Hezekiah, God does tell him that he is about to die. But Hezekiah uses this information. Knowing that God is a man of His word, yet also knowing He is a God of mercy, Hezekiah prays. Now, had Hezekiah not prayed, would he have survived? It seems highly unlikely. And would he have prayed had he not been informed of his impending demise? Hezekiah, being a Godly man, likely did pray on a regular basis, but this particular prayer wouldn't have crossed his mind. Thus, rather than God informing him of his imminent doom being an example of a false prophecy, it appears to be a prompt to pray.
It is worth noting, at this point, that God actually exists outside of time. As Scripture says, a day to Him is as a thousand years, and a thousand years as one day (Psalm 90:4; 2 Peter 3:9). In other words, He exists at all points in time simultaneously. The future, for Him, has already happened, and the past, for Him, has not. But for us?
It goes without saying that, as far as we are concerned, the past is fixed, and the future is not. There are a great many ways things could play out. God knows them all. We do not. God has actualised the ones He desires, but as far as we are concerned, they are mere possibilities. This is why God can make "if/else" statements while still remaining in full control over whether the ifs or the elses occur. He can tell us what will happen if we do, or of we do not repent, not because He doesn't know whether or not we will repent, but because us receiving His if/else statements may influence which occurs. He can say, for example, "...if that nation against whom I have spoken turns from its evil, I will relent of the disaster that I thought to bring upon it" (Jeremiah 18:8), not because He doesn't know whether or not the nation will repent, but because telling them their options will prompt their response.
To be clear, this even makes the danger very real. When God says "I am going to do this", He will do it. You cannot be foolish and think "well actually, He knows what's going to happen anyway, so He'll change His mind in the end." No, He has warned you about what's going to happen, and He knows what you're going to do with that information. Unless you make the wise choice, He will simply maintain the course He was already on, and you are doomed.
But then doesn't this remove the freedom element? After all, God always knows how we will respond regardless, so how are we free? The simple answer is that every decision you make in your freedom, God knows, yet if you had freely chosen a different option, He would know that instead. God, ultimately, is the only one who knows. But what He knows is your choice.
Thus, we see that when God "plays dumb", He is literally playing dumb. He isn't lying. He isn't making educated guesses. He isn't seeking knowledge. He has a goal in mind, and every statement He makes, just as every action He performs, is designed to bring that goal to fruition. This puts the ball in our court. God knows our eternal destiny, but He has set two options before us: Repent, confess Jesus as Lord, and believe in your heart God raised Him from the dead, and your sins will be forgiven, your place in the Kingdom secured, and your relationship with God completely restored. Refuse, and your sins will never be forgiven. You will be paid back every one, a punishment which you can neither bear, nor escape. God knows which option you will choose. Just make sure He knows you chose the right one.