Trying to restore an iPhone backup I took tonight, but can't because when it tries, the startup disk runs out of space. I have other drives, but I can't remove anything else off of the startup disk.
What can I do to restore this backup?
You may want to consider using a utility like OmniDiskSweeper (free) or Daisy Disk ($20) to see if there aren't some unusually large files hiding somewhere, eating up disk space. You can find more apps like these in the answers to How can I free up some disk space?
If your disk is that full and you really can't remove anything else from the startup disk, I would definitely advise upgrading to a larger hard drive. It's generally recommended to keep about 15% of your disk free, and I would certainly advise against getting below 10 GB of free space on your startup disk.
Not having enough disk space on your startup disk can easily cause severe performance problems and more frequent crashing — especially if you completely run out of space.
Other than upgrading your hard drive, you could consider temporarily moving some larger files to another drive until you have enough room for the restore, then, after the restore has completed successfully, you could copy them back to the startup disk.
Depending on what's taking up disk space, it may be worth looking into permanently moving some files to other drives. For example, iTunes and iPhoto libraries, which can often take up a lot of space, can be moved to other drives fairly easily.