First, consider if you have the right backpack. If you're considering chopping off all sorts of parts, you might be using a pack that isn't designed for you. You could sell it off and buy a more minimalist pack, such as one from GoLite (I say GoLite because I'm unfamiliar with other lightweight packs).
With that out of the way, consider what you want to use or carry. If there is no possible reason you would want to strap things to the outside of your pack, feel free to cut off those outside straps completely (the ones on the back that don't compress your pack at all). Most people like something there to attach things to when necessary, however, such as wet clothing, boots, wag bags, etc. There are sometimes one or two little loops of strap on the bottom of the back of your pack. Those are specifically there for mountaineering or ice axes. If you aren't going to be using those, cut them off (though they are good attachment points for other items, if you need it).
If your pack won't be too overloaded, you can probably cut off the strap that goes across the top of your pack (the one underneath the lid, or "brain"). If you roll up the opening enough, there isn't much need for it - though I use mine to hold a rope or other large squishy items.
There are sometimes extra buckles, clips, and straps floating around on your pack. Find what they do, and answer in your head whether they are crucial to the load bearing of your pack or not. If they aren't, then they are convenience items and can be removed at your discretion (melt the ends of the nylon to make them pretty once you cut). If they are compression straps (usually these are on the side of the pack, and are pretty important to the comfort of the pack), don't cut them. You'll be much happier with them still around, despite the ounce of weight. The same goes for straps that hold the pack on you. Shoulder and waist belt straps can be shortened somewhat (carefully!), but don't make them so tiny that you can't make adjustments or put the pack on someone else. If you cut those types of straps, make sure you clean up the cut ends so that they won't slip through whatever buckle system they are installed in.
Finally, be very cautious about trimming any non-strap item. Packs are notoriously difficult to repair if you cut up the wrong thing. Occasionally there are places you can cut off non-strap items, such as attached pockets or cute frills, but know that they might be hiding an important seam, or might be providing much needed waterproofness (as an aside, remember that there really aren't any "waterproof" packs. The material may be waterproof, but the seams aren't, and seam sealing an entire pack is very challenging).