How much are you willing to spend? A forensic recovery outfit could probably do it, but it will cost.
For a DIY job you'll need to be able to boot it and unlock it, so you can use either USB data transfer or USB debugging (depending on what data you need to recover), or run a backup app and copy the backup off. In other words, you'll need to put the motherboard into another phone (or at least connect power and touchscreen to it, but putting it in another phone is the simplest way of doing that). Edit: and this is, of course, assuming that the motherboard is actually functional, which as Scary points out is not a given.