Dx: diabetes inspidus, dry mouth/lips for the last 5 weeks. Drinking tons of water. No other signs of dehydration. How to combat the dry mouth/lips?

Underlying cause. Treat the underlying cause of the di. If this is central di, DDAVP (desmopressin) can help. Otherwise the free water will flow right out of your body.
Lemon drops. Suck on life savers or lemon drops.
Treatment! Why are you not on treatment? There is great treatment for central di and also treatment for nephrogenic di. Treating di will get rid of your dry mouth.
Multi-strategize! You are wise to go after the dry mouth; this causes gum disease, tooth decay and a handful of other oral problems. No one answer will address your needs. Water is a start, find a good lip balm that you like; check out biotene saliva substitutes and similar products; evaluate your diet, supplements & medications for other causative factors that could be changed; stay ahead of the oral health game.