Will we be bored in heaven?
It's not an uncommon question. In fact, a fellow seminary student asked me that very thing. But, the short answer is, no. In fact, I'm persuaded that it will be logically impossible to be bored in heaven. Likewise, it will be logically impossible to experience any good in hell—despite those who hope beer is on tap.
What is heaven?
It's not unusual for many people to conceive of heaven as a place of golden streets, mansions, feasts, the playing of harps, singing and worshiping God for all eternity, etc. In that case, I can understand why some might think that doing that for eternity would likely get old. But, I think imagining what we'll be doing to occupy our time is not helpful. After all, gold streets would probably grow mundane and playing harps just sounds terrible. Instead, I like to reflect on Who's presence we'll be in.
First, it's important to remember who God is. God is the source of all good things (Ps 16:2) and that every good and perfect gift is from above (James 1:17). Consider also that God is perfectly good and the source of moral goodness, He is the locus of justice, mercy, love, and every other good moral attribute.
Next, take account of all of the good in your life, like: moments of joy, times of togetherness and fellowship, the feeling of fulfillment, completeness, purpose, being awestruck by the beauty of nature, being loved and loving others. Reflect on virtuous attributes, like: patience, kindness, forgiveness, generosity, compassion, etc. Now, imagine being in the presence of the source of all those good things. Except, now these things are spilling over in overflowing goodness. To be in the presence of God would mean to experience overflowing joy, the ultimate and complete fulfillment of the purpose of human existence, Whose incomprehensible love draws us into an endlessly joyous pursuit of more of Him. Boredom, however, is not good. Boredom is the lack of fulfillment and purpose. Boredom is loneliness and the absence of loving others. Boredom, therefore, is not a good thing and, so, not anything we would experience in the presence of God.
What is hell?
Hell, like heaven, is also often misconceived. Fanciful conceptions of hell think of it to be something like working in a coal mine. Miserable, tired, dirty, really hot and sweaty, and after a long day you head to the local pub in Hell where they serve terrible beer and you sulk with others in your misery until you make your way back to the coal mine to do it all over again . Other medieval conceptions, even parts of scripture, depict Hell as torture racks, lakes of fire, pain and suffering. That does sound miserable, but that's not hell.
Imagine being separated from the sole source of goodness. That's what Hell is, being separated from God (2 Thes 1:9, Matt 7:23, Luke 16:26) who is the source of all good things. You're not merely separated from any good thing, you're separated from even the possibility of any good thing. Scripture is pretty illustrative of Hell: lakes of fire, gnashing of teeth, darkness, pain and eternal torment. But, I'd say the reality of Hell is likely worse than can be put into words. Hell would be the antithesis of anything good, like: never-ending sorrow and suffering; absolute loneliness, emptiness, and purposelessness; crippling fearfulness; feeling forgotten and abandoned, no reprieve from the misery or torment of a life you chose, forever. What makes Hell even worse, you won't even have the desire for anything good. Meaning, you won't eventually change your mind and cry out, "God, I made a mistake, I want you, I need you in my life!" The rich man in hell didn't (Luke 16:19-31). Why? Because it's the Holy Spirit that draws us to God (John 6:44), Who you also rejected, and from Whom you are also separated. Instead, despite your tormented state, you dig in your heels and continue to reject God. It's a self-inflicted wound that you refuse to let heal. You've rejected the source of all good things and, so, God has given you over to your desires (Rom 1:24). If the source of all good things has been separated from you, no good thing can occur. Thus, it is logically impossible to experience anything good in Hell.