To show that this objection is logically flawed, let's just say that it is true that God doesn't answer prayers. Let's grant the skeptic his first claim, "God doesn't answer prayer. Christians are praying to a God, and they don't receive what it is that they are asking for." My response would take the form of a question, "How do you then conclude that God doesn't exist?" This is a logical fallacy called a non-sequitur; it doesn't follow. Maybe God simply doesn't respond to requests. How is it that you can now deny His existence?
The skeptic will then respond by saying, "Don't you claim to have a God, who hears and answers prayer? Don't you, as a Christian, expect a response? Your Bible says, 'Hear my prayer, O Lord; give ear to my pleas for mercy! In your faithfulness answer me, in your righteousness!' Psalm 143:1 The simple answer to that is yes, God does answer prayer. He answers every prayer, every time. It is answered in one of three ways: "yes, no or wait."
God doesn't promise He will answer every request with a "yes". Many times he says "no" or "later". If you are a parent, do you give your child everything that they ask for? Do you have your reasons when you say no to their request? Is your child sometimes unable to comprehend why it is you are saying no? Similarly, God has His reasons for sometimes not answering prayers in the way we would like Him to. God says, "For my thoughts are not your thoughts, neither are your ways my ways, declares the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts." Isaiah 55:8-9
Even as Christians there are certain things that can hinder our prayers. The biggest reason would be sin. The fact is that many people want to order God around like a magic genie, and have Him do their bidding. Since sin is the biggest hindrance to prayer, let's look at it like this: Unbelief is a sin, and so is pride. So let's take a pride-filled unbeliever and look at his charge against God, "Show yourself to me by performing an undeniable miracle, or appear before me right now - then I will believe in you." Technically, that is a prayer; it's talking to God. The problem is that it is a command to the Creator of the universe, from one of His creations who is filled with pride and arrogance. Who are you to make such demands? What obligation is God under to respond to your commands? Of course He's not going to give you want you want.
It always comes down to a heart issue. "God opposes the proud but gives grace to the humble." 1 Peter 5:5 Those individuals who come to God with a soft heart, who come to Him like a child does to a father, those people will have their prayers answered. Maybe the answer isn't what they wanted to hear, but it does get answered. Like the Scriptures say, Come and hear, all you who fear God, and I will tell what he has done for my soul. I cried to him with my mouth, and high praise was on my tongue. If I had cherished iniquity in my heart, the Lord would not have listened. But truly God has listened; he has attended to the voice of my prayer. Blessed be God, because he has not rejected my prayer or removed his steadfast love from me! Psalms 66:16-20