Sometimes I think about the fact that if I spent all day, every day of my life, reading from sunrise to sunset (starting as soon as I learned how to read, with Dr. Suess, working my way up from there), when I died, there would probably still be thousands of good books (good books, mind you) that I hadn’t read yet. My mind reels.
If I was to live forever, then I would have time to read some trashy books, too, and keep up with subscriptions to Scientific American, The Economist, and National Geographic, plus take a day off now and then to go fishing or something, and I’d still get to read all the good books–and reread 'em as often as the fancy struck. I could learn every language in the world, and read every good book in every language, living and dead.
I’m assuming an eternal life of leisure, of course. If I could live forever, but never graduated from grad school, well, that’d be a whole 'nother kettle of fish. 
To be perfectly serious for a moment, though, I don’t see why one wouldn’t cherish a loved one if the loved one would never die–or if we had to live forever after the someone died. If we were to live forever, I certainly hope that we would spend an eternity growing and changing. The people we loved would probably come and go from our lives, and that’s not a bad thing. And if we could stop loving to avoid the pain of loss, then I think we’d do it even during our brief lives on earth. . . and nobody I know has discovered that secret yet.
Confronting a truly infinite lifetime is a bit difficult for the human imagination, but IMHO the Universe must hold infinite delights. I really do not think it’s true that there’s nothing new under the sun. I think that after a while, we just lose our taste for novelty, and when confronted with a new situation, we are too eager to force it to conform to our old prejudices and preconcieved notions. Anyway, a few billion people with infinite free time would probably be a continuous source of new diversions. The human race has been around for (give or take) 100,000 years, and it took us this long to invent the Nintendo Gamecube, Xtreme sports and “reality” TV. What’s coming next?
But, then, what do I know about infinity? I’m not even thirty yet. 
p.s. If you want to work this from a religious angle, I always figured that part of being in Heaven would be a perfect joy that comes from being with God. Perfect joy could last eternally–and, heck, wouldn’t it have to? If a perfect joy every grew stale, it wouldn’t be perfect, now, would it? But that’s all in the realm of religion, where we can lob absolutes around like dollops of whipped cream. Quite foreign from the human experience . . .