You don't say what type of conifer it was, how big it was or how long you have had it for. As trees go, some conifers are relatively short-lived. Alternatively, it may have died through disease, or through drought. It's hard to say without more information.
If it was a Leylandii, personally I'd throw a party!
If the needles are on a lawn or something like that, I'd definitely rake them up or hoover them up with a blower. I might put them down elsewhere in the garden as a weed suppressant mulch though, especially if I had very alkaline soil (which I do).
christopher2 suggests growing a clematis Montana through the dead branches rather than cutting down & digging out the dead tree. That's a good idea and would work well, if the shape is good, but thinking about some of the dead and dying conifers I've seen in my lifetime, personally I remain to be convinced. Do you know of one you can show us in a photo, christopher2?